This is a summary of the article The Grand Unified Theory of Rogue Waves from Quanta Magazine.
A TL;DR follows.
The mysterious sinking of the MS München has puzzled scientists since 1978. Rumors of a monstrous wave were followed by the discovery of a lifeboat that had been ripped from the vessel. While monstrous waves had been known of mariners for centuries, science did not believe such outliers possible. Until 1995, when a 26-meter rogue wave was recorded on the Draupner oil installation in the Norwegian North Sea.
Two main competing theories attempt to explain the formation of rogue waves. Applied mathematicians focusing on the outcome rather than the formation mechanism found that rogue waves may share a single common characteristic.
The first theory is simply the linear addition of waves. This theory however underestimates the likelihood of rogues. On the other hand, inclusion of triangular and rounded waves in theoretical equations might account for them.
The second theory explains the phenomenon by a nonlinear process of energy leaks between adjacent waves, an effect captured by the nonlinear Schrödinger's equation (NLSE), also known as nonlinear focusing. But ocean waves' characteristics differ from those of tank-generated waves. It might take both mechanisms to account for all observed rogue waves.
A new statistical model could predict the likelihood of any rogue wave. A universal theory of rogue waves requires an understanding of extremely rare events. Not all rare events are necessary: only the least rare suffice. Large deviation theory (LDT) looks for the least rare event possible in a given situation, which is impossible with standard statistics.
So which ocean conditions are required to produce the least unlikely rogue wave?
A numerical model simulated various sea states with specific ranges of heigths and wavelengths. For each sea state, only one particular set of initial conditions ended up forming a rogue wave.
A similar experiment with sets of sea states was then implemented in a one-dimensional physical wave simulator. Only high-level statistical properties were measured. They turned out to be sufficient. Large deviation theory is in essence an optimization problem predicting, in this case, the typical life trajectory of a wave given its initial conditions, and this trajectory is inexorable. The cause of those waves, linear, nonlinear, or both, did not matter.
So initial conditions matter, and neither linear, nor nonlinear theories can accurately predict rogues from them, whereas LDT does. These findings could allow ocean engineers to estimate risks of rogue waves, cutting down the cost of tests required for insurance.
Certain effects are left out from this model, such as currents and winds, because they are not included in the NLSE. The NLSE used in the model can nonetheless be replaced by other wave theories, such as one that lets waves spread freely as they do in the open ocean.
More complete, history-capturing measurements of ocean waves are underway. Two stereoscopic cameras on the upper deck of a sailing vessel measured two weeks of sea surface movements, including those during a cyclone.
Analyzing the ocean surface motion showed nonlinear focusing seems to have caused the large wave and that the LDT model looks promising.
This new universal model marks a small step in the journey towards rogue wave prediction aboard vessels.
TL;DR
While known of mariners for centuries, rogue waves had so far gone unrecorded by scientists, until 1995.
Two main competing wave theories, linear addition and nonlinear focusing, attempted to explain them, but only partially succeeded. Then a proposed universal theory of rogue waves came, made possible thanks to large deviation theory (LDT), which considers the least rare of all rare possible events. A preliminary version of the theory has been used on high-level wave data. Results from both a numerical and a physical wave simulation of rogue waves show that initial conditions are crucial, and that one particular set of initial conditions will cause the most likely rogue wave.
Measurements capturing wave history are underway. According to preliminary results, the LDT model looks promising, paving the way for rogue wave prediciton aboard vessels.
Photo credit: Dave Sandford
