Long nonlinear internal waves

Author Posting. © Annual Reviews, 2006. This article is posted here by permission of Annual Reviews for personal use, not for redistribution. The definitive version was published in Annual Review of Fluid Mechanics 38 (2006): 395-425, doi:10.1146/annurev.fluid.38.050304.092129.Over the past four decades, the combination of in situ and remote sensing observations has demonstrated that long nonlinear internal solitary-like waves are ubiquitous features of coastal oceans. The following provides an overview of the properties of steady internal solitary waves and the transient processes of wave generation and evolution, primarily from the point of view of weakly nonlinear theory, of which the Korteweg-de Vries equation is the most frequently used example. However, the oceanographically important processes of wave instability and breaking, generally inaccessible with these models, are also discussed. Furthermore, observations often show strongly nonlinear waves whose properties can only be explained with fully nonlinear models.KRH acknowledges support from NSF and ONR and an Independent Study Award from the Woods Hole Oceanographic Institution. WKM acknowledges support from NSF and ONR, which has made his work in this area possible, in close collaboration with former graduate students at Scripps Institution of Oceanography and MIT

Three-Dimensional Steady Capillary-Gravity Waves

. Three-dimensional steady capillary-gravity water-waves are studied in this paper. Potential ow of an ideal uid in a layer with nite depth and upper free surface is considered. The existence of these waves is derived through bifurcation processes from the state of rest. The waves are assumed to be periodic in the direction of propagation and just bounded in the transverse direction (modulated periodic travelling waves - MPTW). Restricting the analysis to small amplitude waves, one can reduce the problem to a nite-dimensional reversible and reectionally symmetric dynamical system. Existence and full information about the geometry of the shape of possible crests then follows via normal form analysis and persistence. 1 Introduction Within this \Schwerpunkt DANSE" the project of the two authors of this paper has been mainly concerned with identifying scenarios of socalled dimension -breaking bifurcations for certain free-boundary value problems. These problems arise e.g. when a stea..