Rogue Waves in the Ocean, the Role of Modulational . . .
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equilibrium due to rapidly varying meteorological conditions, or significantly nonuniform environments such as currents or bathymetries [36], or possibly the sudden
appearance of a ship in a wave field [34, 35], can be a significant source of rogue
waves.
The paper is organized as follows: section “Common Theories for Rogue Waves”
gives a summary of common theories on which current rogue wave warning criteria are typically based, section “Case Study on Rogue Waves Through Common
Theories: The Prestige Accident” presents a case study for the involvement of
rogue waves in the Prestige oil tanker accident based on the common theories, section
“Rogue Waves in Non Equilibrium Wave Fields” points out an alternative type of
mechanism for rogue wave generation typically not included in todays warning criteria, and “Conclusions” provides a conclusion.
Common Theories for Rogue Waves
Within linear wave theory (LWT) there are several mechanisms that can provoke
large waves, e.g. spatio-temporal focusing of waves, refraction over uneven depth
and refraction over non-uniform currents. Within LWT, employing the principle of
superposition and the Central Limit Theorem from probability theory, it is anticipated that the resulting distribution of surface elevation is Gaussian (e.g. Pierson,
[46]).
Rogue waves are known to occur more often than anticipated from LWT, this
enhanced occurrence is generally accepted to be due to nonlinearity. There are several known nonlinear mechanisms that can be responsible for this.
Second-order corrections to deep-water waves, static or bound wave nonlinear
corrections in general, are known to provoke small deviation from Gaussian statistics. These corrections were derived for uniform waves by Stokes [50], for deepwater irregular gravity waves by Tick [53] and Longuet-Higgins [29] and further by
Masuda et al. [33]. Static nonlinear corrections to linear wave theory form the basis
for Tayfun-distributions [52].
Starting with the observation that steady uniform waves are unstable to small
perturbations, the Benjamin–Feir instability [3, 4], or more general modulational
instability (MI) was soon recognized as a mechanism that could initiate the generation of extreme waves. It was soon recognized that the cubic nonlinear Schrödinger
(NLS) equation [5, 13, 23, 64] is the simplest nonlinear model that accounts for
this instability mechanism. The MI occurs if the ratio between the steepness of the
uniform wave and the spectral bandwidth of the perturbation is above a threshold.
Soon after the first well-documented observations of rogue waves in the ocean,
e.g. the Draupner wave, it was suggested that the generation of such waves in
the ocean could be explained by weakly nonlinear and narrow-banded models, in
particular by the nonlinear Schrödinger (NLS) equation. Trulsen and Dysthe [56]
argued that it would be an advantage to use a broader-bandwidth modification of the
241
equilibrium due to rapidly varying meteorological conditions, or significantly nonuniform environments such as currents or bathymetries [36], or possibly the sudden
appearance of a ship in a wave field [34, 35], can be a significant source of rogue
waves.
The paper is organized as follows: section “Common Theories for Rogue Waves”
gives a summary of common theories on which current rogue wave warning criteria are typically based, section “Case Study on Rogue Waves Through Common
Theories: The Prestige Accident” presents a case study for the involvement of
rogue waves in the Prestige oil tanker accident based on the common theories, section
“Rogue Waves in Non Equilibrium Wave Fields” points out an alternative type of
mechanism for rogue wave generation typically not included in todays warning criteria, and “Conclusions” provides a conclusion.
Common Theories for Rogue Waves
Within linear wave theory (LWT) there are several mechanisms that can provoke
large waves, e.g. spatio-temporal focusing of waves, refraction over uneven depth
and refraction over non-uniform currents. Within LWT, employing the principle of
superposition and the Central Limit Theorem from probability theory, it is anticipated that the resulting distribution of surface elevation is Gaussian (e.g. Pierson,
[46]).
Rogue waves are known to occur more often than anticipated from LWT, this
enhanced occurrence is generally accepted to be due to nonlinearity. There are several known nonlinear mechanisms that can be responsible for this.
Second-order corrections to deep-water waves, static or bound wave nonlinear
corrections in general, are known to provoke small deviation from Gaussian statistics. These corrections were derived for uniform waves by Stokes [50], for deepwater irregular gravity waves by Tick [53] and Longuet-Higgins [29] and further by
Masuda et al. [33]. Static nonlinear corrections to linear wave theory form the basis
for Tayfun-distributions [52].
Starting with the observation that steady uniform waves are unstable to small
perturbations, the Benjamin–Feir instability [3, 4], or more general modulational
instability (MI) was soon recognized as a mechanism that could initiate the generation of extreme waves. It was soon recognized that the cubic nonlinear Schrödinger
(NLS) equation [5, 13, 23, 64] is the simplest nonlinear model that accounts for
this instability mechanism. The MI occurs if the ratio between the steepness of the
uniform wave and the spectral bandwidth of the perturbation is above a threshold.
Soon after the first well-documented observations of rogue waves in the ocean,
e.g. the Draupner wave, it was suggested that the generation of such waves in
the ocean could be explained by weakly nonlinear and narrow-banded models, in
particular by the nonlinear Schrödinger (NLS) equation. Trulsen and Dysthe [56]
argued that it would be an advantage to use a broader-bandwidth modification of the