Local and global features of resonant third-order interactions amongst surface gravity waves
Keywords: Modified Nonlinear Schrödinger Equation
Nonlinear ocean wave mechanics
OceanWave3D

The formation of 'rogue' or 'freak' waves at the ocean surface is an important consideration for engineering design since such waves represent the worst-case scenario for structures in the ocean. Extreme wave events can be formed by the linear mechanism of dispersion, due to constructive interference between propagating wave components with different frequencies and directions. Nonlinear energy transfers between wave components can, however, alter the shape, kinematics and lifespan of an extreme wave event. The relative importance of linear and nonlinear mechanisms in the formation of extreme wave events remains an active area of interest. This thesis seeks to characterise the local nonlinear interactions which occur during the formation of an isolated group of steep surface gravity waves. We also consider the impact of steepness, finite depth, spreading and the spectral tail on the dominant interactions of an isolated wave group. To perform our investigation, we use a fully-nonlinear potential flow solver, OceanWave3D. We also perform simulations using the Modified Nonlinear Schrödinger (MNLS) equation of Trulsen et al. (2000, Phys. Fluids, vol. 12, pp. 2432-2437) and compare the results with OceanWave3D to assess the fidelity. Lastly, we simulate random seas and assess whether the global spectral changes are consistent with our local observations based on isolated wave groups. If an isolated wave group is narrow-banded, we find that the dominant interactions resemble a third-order degenerate quartet, resulting in energy transfers at ±35° to the spectral peak. As the wave spectrum broadens, the qualitative agreement with a degenerate quartet worsens and we observe energy transfers at ±55° to the spectral peak. We find that the the ±35° and ±55° energy transfers are both suppressed by finite depth but the ±35° interactions are more severely suppressed, effectively disappearing at kd=1.36. In contrast, the ±55° interactions are more resilient to a reduction in depth and remain prominent at kd=1.36. An isolated wave event with high directional spreading will focus for a comparatively shorter amount of time due to linear dispersion, reducing energy transfers in an integral sense. We find that high directional spreading also tends to reduce the rate of energy transfer between wave components. Similarly, we find that truncation of the tail of the spectrum tends to augment the rates of energy transfer between wave components, accompanied by rapid redevelopment of the spectral tail. Thus, we concluded that directional spreading and the high-wavenumber tail of the spectrum play an important role in establishing a form of spectral equilibrium that reduces nonlinear energy transfers for an isolated wave event. Our comparison between the MNLS equation and OceanWave3D reveals that the MNLS equation generally captures the formation of an isolated wave group of surface gravity waves. However, the equation performs better if the linear dispersion relation is exact, rather than truncated at the fifth-order. The exact version of the dispersion operator also allows the arbitrary-depth linear dispersion relation to be included in the equation. We find that the classic MNLS equation of Trulsen et al. (2000) performs well at finite depths (kd) between 5.59 and 1.36 if the exact dispersion operator is combined with the the arbitrary-depth linear dispersion relation and the bottom boundary condition is applied at finite depth. Despite the existence of several intermediate-depth MNLS formulations, our results suggest that modification of the coefficients to the nonlinear terms is not required within this range of depths. Lastly, our simulations of random seas supports our observations based on isolated wave groups. We find that truncation of the spectral tail also tends to augment the energy transfers of a random sea, accompanied by redevelopment of spectral tail and augmented values of the free-surface kurtosis. Thus, we conclude that the spectral tail also plays an important role in establishing the spectral equilibrium of random seas and care should be taken when truncating the tail for the purposes of simulation/experiment. Our results suggest that the accumulation of local spectral distortions caused by individual wave groups could play an important role in determining the global spectral evolution of a random sea.