Rapid spectral evolution of steep surface wave groups with directional spreading

We have investigated steep three-dimensional surface gravity wave groups formed by dispersive focusing using a fully nonlinear potential flow solver. We find that third-order resonant interactions result in rapid energy transfers to higher wavenumbers and reduced directional spreading during focusing, followed by spectral broadening during defocusing, forming steep wave groups with augmented kinematics and a prolonged lifespan.

If the wave group is initially narrow-banded, quasi-degenerate interactions arise, characterised by energy transfers along the resonance angle, of the Phillips 'figure-of-eight' loop. Spectral broadening due to the quasi-degenerate interactions facilitates non-degenerate interactions, characterised by oblique energy transfers at approximately to the spectral peak.

We consider the influence of steepness, finite depth, directional spreading and the high-wavenumber tail on spectral evolution. Steepness is found to augment both the quasi-degenerate and non-degenerate interactions similarly. However, a reduction in depth is found to weaken the quasi-degenerate interactions more severely than the non-degenerate interactions.

We observe that increased directional spreading reduces spectral evolution, partially because wave groups with more spreading focus for a shorter duration due to linear dispersion. However, we also find that directional spreading reduces the peak rates of energy transfer. Inclusion of the high-wavenumber tail of the Joint North Sea Wave Project spectrum further reduces rates of energy transfer compared with a Gaussian wavenumber spectrum.

Thus, directional spreading and the high-wavenumber tail may be integral to a form of spectral equilibrium that reduces rapid energy transfers during a steep wave event.