Breather Rogue Waves in Random Seas

Rogue or freak waves are extreme wave events that have heights exceeding 8 times the standard deviation of surrounding waves and emerge, for instance, in the ocean as well as in other physical dispersive wave guides, such as in optical fibers. One effective and convenient way to model such an extreme dynamics in laboratory environments within a controlled framework as well as for short process time and length scales is provided through the breather formalism. Breathers are pulsating localized structures known to model extreme waves in several nonlinear dispersive media in which the initial underlying process is assumed to be narrow banded. On the other hand, several recent studies suggest that breathers can also persist in more complex environments, such as in random seas, beyond the attributed physical limitations. In this work, we study the robustness of the Peregrine breather (PB) embedded in Joint North Sea Wave Project (JONSWAP) configurations using fully nonlinear hydrodynamic numerical simulations in order to validate its practicalness for ocean engineering applications. We provide a specific range for both the spectral bandwidth of the dynamical process as well as the background wave steepness and, thus, quantify the applicability of the PB in modeling rogue waves in realistic oceanic conditions. Our results may motivate analogous studies in fields of physics such as optics and plasma to quantify the limitations of exact weakly nonlinear models, such as solitons and breathers, within the framework of the fully nonlinear governing equations of the corresponding medium.

Figure 1

Configuration of the numerical wave tank employed in the numerical study that includes similar features as physical wave tanks, namely, a wave generator and a damping zone.

Figure 2

Model validation by comparison of Peregrine breather temporal profiles for $a=0.01\mathrm{m}$ and $ϵ=0.1163$ for theoretical prediction (dashed red lines), laboratory observation (solid blue lines), MNLS simulation (dash-dot purple lines), and ESBI simulations (dot black lines) at (a) $x=1.1\mathrm{m}$ and (b) maximal compression $x=9.1\mathrm{m}$.

Figure 3

Evolution of the embedded PB in a JONSWAP configuration for $ϵ=0.0873$, $H_s/ϵ=2$, and $\gamma =1$ (a) $T=0$, (b) $T=T_f-3T_p$, (c) $T=T_f$, (d) $T=T_f+3T_p$.

Figure 4

Free-surface profile of a Peregrine-type wave embedded in a JONSWAP configuration for the case $H_s/ϵ=1.333$ and $\gamma =1$ for (a) free-surface spatial distribution and (b) corresponding temporal wave profiles.

Figure 5

Average error versus standard deviation error for four cases, as described in the figure’s legend.

Figure 6

$\mathrm{avg}(I_A)\pm \mathrm{std}(I_A)$ versus JONSWAP peakedness parameter $\gamma$ for $H_s/ϵ=1$.

Figure 7

$\mathrm{avg}(I_A)-\mathrm{std}(I_A)$ versus JONSWAP peakedness parameter $\gamma$ for the cases described in the figure’s legend.

Figure 8

$\max (H_s/ϵ)$ as a function of JONSWAP peakedness parameter $\gamma$.