T. T. Janssen

Nonlinear Wave Statistics in a Focal Zone

Abstract In this paper, the combined effects of refraction and nonlinearity on the evolution of ocean surface wave statistics are considered and possible implications for the likelihood of extreme waves, also known as freak or rogue waves, are examined. A frequency-angular spectrum model is derived that accounts for cubic nonlinear dynamics and weak lateral homogeneity of the medium. Through Monte Carlo simulations, the evolution of wave statistics in freely developing waves, waves over an opposing shearing current, and waves refracted over an isolated topographical feature is modeled. The simulations show that freely developing, directionally spread wave fields generally maintain near-Gaussian statistics, which was also found in earlier model studies. However, the enhanced nonlinearity caused by the refractive focusing of narrowband wave fields can result locally in strongly non-Gaussian statistics and an associated increased likelihood of extreme wave events.

Stochastic modeling of coherent wave fields over variable depth

Refractive focusing of swell waves can result in fast-scale variations in the wave statistics because of wave interference, which cannot be resolved by stochastic wave models based on the radiative transport equation. Quasi-coherent statistical theory does account for such statistical interferences and the associated wave inhomogeneities, but the theory has thus far been presented in a form that appears incompatible with models based on the radiative transfer equation (RTE). Moreover, the quasi-coherent theory has never been tested against field data, and it is not clear how the coherent information inherent to such models can be used for better understanding coastal wave and circulation dynamics. This study therefore revisits the derivation of quasi-coherent theory to formulate it into a radiative transport equation with a forcing term that accounts for the inhomogeneous part of the wave field. This paper shows how the model can be nested within (or otherwise used in conjunction with) quasi-homogeneous wave models based on the RTE. Through comparison to laboratory data, numerical simulations of a deterministic model, and field observations of waves propagating over a nearshore canyon head, the predictive capability of the model is validated. The authors discuss the interference patterns predicted by the model through evaluation of a complex cross-correlation function and highlight the differences with quasi-homogeneous predictions. These results show that quasi-coherent theory can extend models based on the RTE to resolve coherent interference patterns and standing wave features in coastal areas, which are believed to be important in nearshore circulation and sediment transport.

Nonlinear Wave Kinematics near the Ocean Surface

AbstractEstimation of second-order, near-surface wave kinematics is important for interpretation of ocean surface remote sensing and surface-following instruments, determining loading on offshore structures, and understanding of upper-ocean transport processes. Unfortunately, conventional wave theories based on Stokes-type expansions do not consider fluid motions at levels above the unperturbed fluid level. The usual practice of extrapolating the fluid kinematics from the unperturbed free surface to higher points in the fluid is generally reasonable for narrowband waves, but for broadband ocean waves this results in dramatic (and nonphysical) overestimation of surface velocities. Consequently, practical approximations for random waves are at best empirical and are often only loosely constrained by physical principles. In the present work, the authors formulate the governing equations for water waves in an incompressible and inviscid fluid, using a boundary-fitted coordinate system (i.e., sigma or s coordi...