Linear and nonlinear rogue wave statistics in the presence of random currents

Abstract

We review recent progress in modelling the probability distribution of wave heights in the deep ocean as a function of a small number of parameters describing the local sea state. Both linear and nonlinear mechanisms of rogue wave formation are considered. First, we show that when the average wave steepness is small and nonlinear wave effects are subleading, the wave height distribution is well explained by a single 'freak index' parameter, which describes the strength of (linear) wave scattering by random currents relative to the angular spread of the incoming random sea. When the average steepness is large, the wave height distribution takes a very similar functional form, but the key variables determining the probability distribution are the steepness, and the angular and frequency spread of the incoming waves. Finally, even greater probability of extreme wave formation is predicted when linear and nonlinear effects are acting together.

Quodons in Mica

“…Among those, there are coherent phenomena related to branching of waves, the onset of caustic areas as well as rogue wave formation. Of particular interest are phenomena related to electron flow in a two dimensional electron gas [20,32], transport properties of semiconductors [20,32], ocean waves [37], linear and nonlinear light propagation in random fibers [24,28,29], sound wave propagation [5,34,35], microwave devices [4,9], resonance in nonlinear optical cavities [21] and light propagation through random refractive index media [12,18,27,30]. Many of these cases can be analyzed mathematically using a unified framework that provides results valid in quite different circumstances.…”

Systematic study of rogue wave probability distributions in a fourth‐order nonlinear Schrödinger equation

“…Since we still assume a narrow spectrum and a small angular spread, the envelope evolves slowly in space and time, on the scale of the mean wavelength and mean wave period, respectively. In analogy with the situation for linear ray dynamics in the presence of random currents [Ying et al, 2011], in the neighborhood of any spacetime pointr; t ð Þ, we have a wave intensity proportional to…”

“…[30] Typical results are represented by dashed or dotted curves in Figure 2, where we fix Dk/k 0 = 0.1 and Dq = 2.6 (the values of Dq required to see very strong effects from nonlinear focusing are typically smaller than those needed to observe significant deviations from Rayleigh by linear scattering [Ying et al, 2011]). The cumulative probability distribution of the wave height 2H, in units of the significant wave height H s , is shown for three nonzero values of the wave steepness ɛ.…”

Publisher's Note:N-order bright and dark rogue waves in a resonant erbium-doped fiber system [Phys. Rev. E86, 066603 (2012)]

“…In recent years, in addition to solitons in different optical systems, the study of rogue waves have also attracted considerable interest because of their potential applications in different branches of physics including oceanography [21][22][23][24], which occurs due to either modulation instability [25][26][27][28][29][30][31], or random initial condition [24,32]. The first order rogue wave is most likely to appear as a single peak hump with two caves in a plane with a nonzero boundary.…”