Rogue Waves in Random Sea States: An Experimental Perspective

Abstract

Despite being rare events, rogue waves have been recorded in the ocean. Here the current knowledge on wave statistics and the probability of occurrence of rogue waves is revisited from an experimental perspective. Starting from the instability of uniform wave packets to side band perturbations, the most accredited generating mechanism, the appearance of rogue waves in random wave fields is discussed. As an initial condition, unidirectional wave propagation is considered. Under these circumstances, wave instability results in a substantial deviation from Gaussian statistics. Directional spreading of wave energy, which characterizes realistic oceanic waves, attenuates the effect of wave instability weakening non-Gaussian properties. It is demonstrated, however, that the interaction between waves and an opposing current can sometimes act as a catalyst for modulational instability. This triggers the formation of rogues waves even in directional sea states, where rogue waves are the least expected.

Appendix

The analytical form of the coupling coefficients K _ijlm ⁺ and K _ijlm ⁻ of the second order theory are as follows:

$$ \displaystyle\begin{array}{rcl} K_{ijlm}^{+} = [D_{ ijlm}^{+} - (k_{ i}k_{j}\cos (\theta _{l} -\theta _{m}) - R_{i}R_{j})](R_{i}R_{j})^{-1/2} + (R_{ i} + R_{j})& &{}\end{array}$$

(4)

$$\displaystyle\begin{array}{rcl} K_{ijlm}^{-} = [D_{ ijlm}^{-}- ((k_{ i}k_{j}\cos (\theta _{l} -\theta _{m}) + R_{i}R_{j})](R_{i}R_{j})^{-1/2} + (R_{ i} + R_{j})& &{}\end{array}$$

(5)

where

$$\displaystyle\begin{array}{rcl} D_{ijlm}^{+}& =& \frac{(\sqrt{R_{i}} + \sqrt{R_{j}})\{\sqrt{R_{i}}(k_{j}^{2} - R_{ j}^{2}) + \sqrt{R_{ j}}(k_{i}^{2} - R_{ i}^{2})\}} {(\sqrt{R_{i}} + \sqrt{R_{j}})^{2} - k_{ijlm}^{+}\tanh (k_{ijlm}^{+}h)} + \\ & & \frac{2(\sqrt{R_{i}} + \sqrt{R_{j}})^{2}(k_{i}k_{j}\cos (\theta _{l} -\theta _{m}) - R_{i}R_{j})} {(\sqrt{R_{i}} + \sqrt{R_{j}})^{2} - k_{ijlm}^{+}\tanh (k_{ijlm}^{+}h)} {}\end{array}$$

(6)

$$\displaystyle\begin{array}{rcl} D_{ijlm}^{-}& =& \frac{(\sqrt{R_{i}} -\sqrt{R_{j}})\{\sqrt{R_{j}}(k_{i}^{2} - R_{ i}^{2}) -\sqrt{R_{ i}}(k_{j}^{2} - R_{ j}^{2})\}} {(\sqrt{R_{i}} -\sqrt{R_{j}})^{2} - k_{ijlm}^{-}\tanh (k_{ijlm}^{-}h)} + \\ & & \frac{2(\sqrt{R_{i}} -\sqrt{R_{j}})^{2}(k_{i}k_{j}\cos (\theta _{l} -\theta _{m}) + R_{i}R_{j})} {(\sqrt{R_{i}} -\sqrt{R_{j}})^{2} - k_{ijlm}^{-}\tanh (k_{ijlm}^{-}h)} {}\end{array}$$

(7)

$$\displaystyle\begin{array}{rcl} k_{ijlm}^{-}& =& \sqrt{k_{ i}^{2} + k_{j}^{2} - 2k_{i}k_{j}\cos (\theta _{l} -\theta _{m})}{}\end{array}$$

(8)

$$\displaystyle\begin{array}{rcl} k_{ijlm}^{+}& =& \sqrt{k_{ i}^{2} + k_{j}^{2} + 2k_{i}k_{j}\cos (\theta _{l} -\theta _{m})}{}\end{array}$$

(9)

$$\displaystyle\begin{array}{rcl} R_{i} =\omega _{ i}^{2}/g& &{}\end{array}$$

(10)