Dynamics of three-dimensional gravity-capillary solitary waves in deep water

A model equation for gravity-capillary waves in deep water is proposed. This model is a quadratic approximation of the deep water potential flow equations and has wavepacket-type solitary wave solutions. The model equation supports line solitary waves which are spatially localized in the direction of propagation and constant in the transverse direction, and lump solitary waves which are spatially localized in both directions. Branches of both line and lump solitary waves are computed via a numerical continuation method. The stability of each type of wave is examined. The transverse instability of line solitary waves is predicted by a similar instability of line solitary waves in the nonlinear Schrödinger equation. The spectral stability of lumps is predicted using the waves' speed energy relation. The role of wave collapse in the stability of these waves is also examined. Numerical time evolution is used to confirm stability predictions and observe dynamics, including instabilities and solitary wave collisions

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

The propagation of internal solitary waves over variable topography in a horizontally two-dimensional framework

This paper presents a horizontally two-dimensional theory based on a variable-coefficient Kadomtsev-Petviashvili equation, which is developed to investigate oceanic internal solitary waves propagating over variable bathymetry, for general background density stratification and current shear. To illustrate the theory, a typical monthly averaged density stratification is used for the propagation of an internal solitary wave over either a submarine canyon or a submarine plateau. The evolution is essentially determined by two components, nonlinear effects in the main propagation direction and the diffraction modulation effects in the transverse direction. When the initial solitary wave is located in a narrow area, the consequent spreading effects are dominant, resulting in a wave field largely manifested by a significant diminution of the leading waves, together with some trailing shelves of the opposite polarity. On the other hand, if the initial solitary wave is uniform in the transverse direction, then the evolution is more complicated, though it can be explained by an asymptotic theory for a slowly varying solitary wave combined with the generation of trailing shelves needed to satisfy conservation of mass. This theory is used to demonstrate that it is the transverse dependence of the nonlinear coefficient in the Kadomtsev-Petviashvili equation rather than the coefficient of the linear transverse diffraction term that determines how the wave field evolves. The Massachusetts Institute of Technology (MIT) general circulation model is used to provide a comparison with the variable-coefficient Kadomtsev-Petviashvili model, and good qualitative and quantitative agreements are found

Physical Mechanisms of the Rogue Wave Phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin-Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey - Stewartson system, the Korteweg - de Vries equation, the Kadomtsev - Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon