Nonlinear water waves: introduction and overview

Abstract

For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme ‘Nonlinear water waves’ that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results. This article is part of the theme issue ‘Nonlinear water waves’.

Rotational waves generated by current‐topography interaction

“…A good source of recent articles is found in the theme volume edited by Constantin. 4 Constant vorticity has an impact, for example, on the formation of a critical layer beneath traveling waves. 5 This critical layer has a Kelvin cat eye structure together with the respective formation of critical points, as numerically illustrated by the authors in Ref.…”

“…22 As in the flat bottom case, to solve the present system of differential equations we need the initial conditions Y( , 0) and Φ( , 0). Equation 30 4 indicates that we also need the initial value Ψ( , 0). Therefore, we present the steps in obtaining Ψ( , ) from Φ( , ).…”

“…These include numerical studies with traveling waves, such as that by Vasan and Oliveras as well as by Ribeiro et al Applied analysis researchers recently made efforts in understanding the impact of constant vorticity on nonlinear traveling wave solutions. A good source of recent articles is found in the theme volume edited by Constantin . Constant vorticity has an impact, for example, on the formation of a critical layer beneath traveling waves .…”

“…While more general vorticity distributions may be more realistic, the respective mathematical modeling, as well as theoretical access, are much harder than in the constant vorticity case. In his introduction, Constantin calls attention that “constant vorticity is representative when the waves are long compared with the water depth since in this case it is the existence of a nonzero mean vorticity that is important rather than its specific distribution. In this study, we consider the long wave (weakly dispersive) regime.…”

Forecasting Hurricane-forced Significant Wave Heights using the Long Short-Term Memory Network in the Caribbean Sea

“…Relevant for the current discussion, these methods are also finding use in weather and climate modelling (Kashinath et al, 2021). Considering the large physical complexities in wave evolution under TC forcing (Tamizi et al, 2021), and the many nonlinearities that govern crucial processes (Yim et al, 2017;Constantin, 2018;Sharifineyestani and Tahvildari, 2021), incorporating physics-informed, or knowledge-guided machine learning should, respectively, improve and lengthen forecast efficacy and horizons.…”