The transformation of an interfacial solitary wave of elevation at a bottom step

Abstract. In this paper we study the transformation of an internal solitary wave at a bottom step in the framework of two-layer flow, for the case when the interface lies close to the bottom, and so the solitary waves are elevation waves. The outcome is the formation of solitary waves and dispersive wave trains in both the reflected and transmitted fields. We use a two-pronged approach, based on numerical simulations of the fully nonlinear equations using a version of the Princeton Ocean Model on the one hand, and a theoretical and numerical study of the Gardner equation on the other hand. In the numerical experiments, the ratio of the initial wave amplitude to the layer thickness is varied up one-half, and nonlinear effects are then essential. In general, the characteristics of the generated solitary waves obtained in the fully nonlinear simulations are in reasonable agreement with the predictions of our theoretical model, which is based on matching linear shallow-water theory in the vicinity of a step with solutions of the Gardner equation for waves far from the step.

[1]  T. Kakutani,et al.  Solitary Waves on a Two-Layer Fluid , 1978 .

[2]  V. Vlasenko,et al.  Three‐dimensional shoaling of large‐amplitude internal waves , 2007 .

[3]  K. Helfrich.  Internal solitary wave breaking and run-up on a uniform slope , 1992, Journal of Fluid Mechanics.

[4]  L. Redekopp,et al.  The Fission and Disintegration of Internal Solitary Waves Moving over Two-Dimensional Topography , 1978 .

[5]  R. Grimshaw,et al.  Fission of a weakly nonlinear interfacial solitary wave at a step , 2008 .

[6]  Y. Kanarska,et al.  A non-hydrostatic numerical model for calculating free-surface stratified flows , 2003 .

[7]  Chen-Yuan Chen,et al.  Laboratory observations on internal solitary wave evolution on steep and inverse uniform slopes , 2007 .

[8]  P. Mignerey,et al.  Nonlinear internal waves in the South China Sea: Observation of the conversion of depression internal waves to elevation internal waves , 2003 .

[9]  D. Bogucki,et al.  A mechanism for sediment resuspension by internal solitary waves , 1999 .

[10]  V. Klemas,et al.  Satellite observation of internal solitary waves converting polarity , 2003 .

[11]  R. Grimshaw,et al.  Generation of large-amplitude solitons in the extended Korteweg-de Vries equation. , 2002, Chaos.

[12]  Efim Pelinovsky,et al.  Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface , 2002 .

[13]  K. Helfrich,et al.  On long nonlinear internal waves over slope-shelf topography , 1986, Journal of Fluid Mechanics.

[14]  Atle Jensen,et al.  Properties of large-amplitude internal waves , 1999, Journal of Fluid Mechanics.

[15]  Shoaling solitary internal waves: on a criterion for the formation of waves with trapped cores , 2003, Journal of Fluid Mechanics.

[16]  Kuang-An Chang,et al.  Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle , 2001 .

[17]  K. Hutter,et al.  Generation of second mode solitary waves by the interaction of a first mode soliton with a sill , 2001 .

[18]  R. Grimshaw,et al.  Modelling Internal Solitary Waves in the Coastal Ocean , 2007 .

[19]  Chen-Yuan Chen,et al.  An investigation on internal solitary waves in a two-layer fluid: Propagation and reflection from steep slopes , 2007 .

[20]  R. Grimshaw,et al.  Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves , 2004 .

[21]  K. Lamb,et al.  A numerical investigation of solitary internal waves with trapped cores formed via shoaling , 2002, Journal of Fluid Mechanics.

[22]  P. Holloway,et al.  A Nonlinear Model of Internal Tide Transformation on the Australian North West Shelf , 1997 .

[23]  Dan E. Kelley,et al.  Evolution of a shoaling internal solitary wavetrain , 2007 .

[24]  K. Hutter,et al.  Baroclinic Tides: Theoretical Modeling and Observational Evidence , 2005 .

[25]  K. Lamb,et al.  Sediment resuspension mechanisms associated with internal waves in coastal waters , 2008 .

[26]  P. Holloway,et al.  A model of suspended sediment transport by internal tides , 2001 .

[27]  V. Klemas,et al.  Nonlinear evolution of ocean internal solitons propagating along an inhomogeneous thermocline , 2001 .

[28]  G. Mellor.  An Equation of State for Numerical Models of Oceans and Estuaries , 1991 .

[29]  Roberto Camassa,et al.  Fully nonlinear internal waves in a two-fluid system , 1996, Journal of Fluid Mechanics.

[30]  P. Liu,et al.  A numerical study of the evolution of a solitary wave over a shelf , 2001 .

[31]  Andrea Prosperetti,et al.  Physics of Fluids , 1997 .

[32]  Efim Pelinovsky,et al.  A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone , 1999 .

[33]  Cheng-Wu Chen,et al.  GENERATION OF INTERNAL SOLITARY WAVE BY GRAVITY COLLAPSE , 2007 .

[34]  Pengzhi Lin,et al.  A numerical study of solitary wave interaction with rectangular obstacles , 2004 .

[36]  Dominique P. Renouard,et al.  Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle , 1987, Journal of Fluid Mechanics.

[37]  Kuang-An Chang,et al.  Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle: Part II: Cnoidal waves , 2001 .

[38]  Antony K. Liu,et al.  Internal solitons in the northeastern south China Sea. Part I: sources and deep water propagation , 2004, IEEE Journal of Oceanic Engineering.