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A Large Deviation Theory Approach to Rogue Waves

By Jillian Kunze

Figure 1. Diagram of the wave flume at the Norwegian Marine Technology Research Institute.
Rogue waves are mysterious and extreme events, in which the ocean’s surface reaches an unusually high elevation in the form of a large wave. Unlike tsunamis, which are known to be caused by earthquakes on the ocean floor, these sudden waves’ exact source remains a contentious subject. “The mechanisms of rogue waves are not fully understood, which has prompted ongoing debate over the last few decades,” said Tobias Grafke of the University of Warwick. The best understanding in the field at present is that rogue waves are caused by nonlinear amplification out of a background of smaller waves. However, the relevant probability density function is unknown. In a minisymposium presentation at the 2021 SIAM Conference on Computational Science and Engineering, which is taking place virtually this week, Grafke described his work on modeling rogue wave instances using large deviation theory. He specifically aimed to estimate the tail in the distribution of rogue wave heights, hoping to formulate a mathematical theory that would apply somewhat universally. 

This study was the first time that Grafke based his normally solely theoretical work on experimental data. He collaborated with the Norwegian Marine Technology Research Institute to carry out rogue wave experiments on their 260-meter wave flume (see Figure 1). In the experiment, a wave maker would generate a random signal based on the JONSWAP observational spectrum — a wave spectrum based on data from the Joint North Sea Wave Observation Project. This wave spectrum is parametrized by wind speed and a number of other nautical terms, but can be reduced to 2 significant parameters: the signature wave height and the degree of nonlinearity. 

Figure 2. Paths of a damped pendulum. The colors show possible paths of the pendulum around and between the two stable fixed points from many iterations of the simulation. The white line shows the optimal trajectory between the two fixed points, which goes through the saddle point. Note that there is not a very large distribution of paths around that saddle point.
By pushing back and forth in the water, the wave maker created a planar wave that propagated down the length of the flume. 19 probes distributed along the flume then recorded the elevation of the water as the wave traveled, specially noting when the height exceeded a certain high threshold. By recording the time that a certain probe was triggered as an instance of a rogue wave, Grafke was able to build up a database of extreme events and their precursors. He could track the wave backwards in space by looking at measurements taken previously by other probes, enabling him to investigate the precursor wave packages that would later develop into rogue waves. Using this database of rogue wave instances, Grafke was able to compute what the average extreme event looked like, as well as the standard deviation. He found that the deviation around important wave packages with larger heights was fairly small — a lot of the instances of rogue waves looked very similar. This led him to believe that rogue waves could be explained by large deviation theory.

Figure 3. Rogue wave events in the wave flume compared with instanton theory. The solid colored line shows the mean from the experiment, the colorful packet shows the standard deviation, and the black line shows the trajectory determined by instanton theory.
Grafke then delved into a more theory-driven mathematical explanation of large deviation theory, which can also be called instanton theory. “The way rare events occur is often predictable—it is dominated by the least unlikely scenario—which is the essence of large deviation theory,” Grafke said. To help develop the intuition behind large deviation theory, he presented a simple example of a damped pendulum in two dimensions with noise. One fixed point occurred when the pendulum pointed straight down at \(x=0\); when excited, the pendulum would either move slightly and return to \(0\), or make a full revolution and settle another the fixed point at \(x = 2 \pi\). Between \(x = 0\) and \(x = 2 \pi\) was a saddle point, where the pendulum pointed straight up instead of straight down. In the limit of zero noise, it is possible to rigorously predict what the transition path through that saddle point will look like.

One can estimate the probability of the pendulum transitioning from one stable point to another using a minimization problem. This minimizer—which is also called the instanton, based on similar techniques in quantum field theory—can be found based on the set of trajectories that transition between the two points. Knowledge of the optimal trajectory then reveals many details about the physical processes that are occurring, such as the probability of the event, the most likely occurrence, and the optimal fluctuation. This allows one to make predictions, explore possible causes of physical scenarios, and find the most effective way to force the event. As this scenario is cheap to simulate, Grafke was able to do so a large number of times in order to find the distribution of possible paths (see Figure 2). This revealed that the in the in-between region of the saddle point, the many stochastic trajectories all look fairly similar to the optimal path.

Figure 4. Estimates of the likelihood of certain wave heights in rough seas and high seas on different timescales. The dots show the results from a Monte Carlo simulation, and the solid lines show the results from large deviation theory. Large deviation theory does well at predicting the tails of the distribution.

Grafke then reached the crux of his presentation: combing large deviation theory with the rogue wave measurements. After modeling the surface of the water with the nonlinear Schrödinger equation, he had all the ingredients necessary to compute the instanton and find the probability of exceeding the rogue wave trajectory. Figure 3 demonstrates the comparison between experimental measurements of extreme events in the wave flume and instanton theory; each rogue wave event matches fairly closely to the related instanton solution. The instanton theory was also able to interpolate between the linear and Peregrine, or nonlinear, regimes. 

Based on these calculations, Grafke was able to estimate the likelihoods of certain wave heights (see Figure 4). In comparing Monte Carlo simulations with his results, he found that he had attained his original goal: the comparison revealed that large deviation theory can nicely predict the tails of the rogue wave distribution. Grafke concluded his talk by noting that he was able to show experimental evidence for hydrodynamic instantons, as he found an instanton solution to closely resemble each rogue wave event that was recorded in the wave flume. By including linear theory as a limiting case and the Peregrine soliton in the nonlinear limit, Grafke was able to create a unified description for rogue waves.

  Jillian Kunze is the associate editor of SIAM News.