Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-19T07:11:29.363Z Has data issue: false hasContentIssue false

The evolution of second mode internal solitary waves over variable topography

Published online by Cambridge University Press:  11 December 2017

C. Yuan
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
R. Grimshaw*
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
E. Johnson
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
*
Email address for correspondence: r.grimshaw@ucl.ac.uk

Abstract

A study of the propagation of a mode-2 internal solitary wave over a slope-shelf topography is presented. The methodology is based on a variable-coefficient Korteweg–de Vries (vKdV) equation, using both analysis and numerical simulations, and simulations using the MIT general circulation model (MITgcm). Two configurations are considered. One is a mode-2 internal solitary wave propagating up the slope, from one three-layer system to another three-layer system. Depending on the height of the shelf, which determines the variation of the nonlinear coefficient of the vKdV equation, this can be classified into two cases. First, the case of a polarity change, in which the coefficient of the quadratic nonlinear term changes sign at a certain critical point on the slope, and second, the case with no such polarity change. In both these cases there is a small transfer of energy from the mode-2 wave to mode-1 waves. The other configuration is when the lower layer in the three-layer system goes to zero at a transition point on the slope, and beyond that point, there is a two-layer fluid system. A mode-2 internal solitary wave propagating up the slope cannot exist past this transition point. Instead it is extinguished and replaced by a mode-1 bore and trailing wave packet which moves onto the shelf.

Type
JFM Papers
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akylas, T. R. & Grimshaw, R. H. J. 1992 Solitary internal waves with oscillatory tails. J. Fluid Mech. 242, 279298.CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Courier Corporation.Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29 (3), 593607.Google Scholar
Farmer, D. M. & Smith, J. D. 1980 Tidal interaction of stratified flow with a sill in knight inlet. Deep Sea Res. A. Oceanogr. Res. 27 (3–4), 239IN5247246IN10254.Google Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Elsevier.Google Scholar
Grimshaw, R. 1981 Evolution equations for long, nonlinear internal waves in stratified shear flows. Stud. Appl. Maths 65 (2), 159188.CrossRefGoogle Scholar
Grimshaw, R. 2006 Internal solitary waves in a variable medium. GAMM-Mitteilungen 30 (1), 96109.Google Scholar
Grimshaw, R., Pelinovsky, E. & Talipova, T. 2007 Modelling internal solitary waves in the coastal ocean. Surv. Geophys. 28 (4), 273298.Google Scholar
Grimshaw, R., Pelinovsky, E., Talipova, T. & Kurkin, A. 2004 Simulation of the transformation of internal solitary waves on oceanic shelves. J. Phys. Oceanogr. 34 (12), 27742791.CrossRefGoogle Scholar
Grimshaw, R., Pelinovsky, E., Talipova, T. & Kurkina, O. 2010 Internal solitary waves: propagation, deformation and disintegration. Nonlinear Process. Geophys. 17 (6), 633649.Google Scholar
Grimshaw, R. & Yuan, C. 2016 The propagation of internal undular bores over variable topography. Phys. D 333, 200207.Google Scholar
Guo, C. & Chen, X. 2012 Numerical investigation of large amplitude second mode internal solitary waves over a slope-shelf topography. Ocean Model. 42, 8091.Google Scholar
Holloway, P. E., Pelinovsky, E. & Talipova, T. 1999 A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone. J. Geophys. Res. 104 (C8), 1833318350.Google Scholar
Huang, X., Chen, Z., Zhao, W., Zhang, Z., Zhou, C., Yang, Q. & Tian, J. 2016 An extreme internal solitary wave event observed in the northern south China sea. Sci. Rep. 6, 30041.CrossRefGoogle ScholarPubMed
Konyaev, K. V., Sabinin, K. D. & Serebryany, A. N. 1995 Large-amplitude internal waves at the mascarene ridge in the indian ocean. Deep Sea Res. I 42 (11–12), 2075208320812091.Google Scholar
Lamb, K. G. 2007 Energy and pseudoenergy flux in the internal wave field generated by tidal flow over topography. Cont. Shelf Res. 27 (9), 12081232.Google Scholar
Leith, C. E. 1996 Stochastic models of chaotic systems. Phys. D 98 (2–4), 481491.CrossRefGoogle Scholar
Liu, A. K., Su, F.-C., Hsu, M.-K., Kuo, N.-J. & Ho, C.-R. 2013 Generation and evolution of mode-two internal waves in the south china sea. Cont. Shelf Res. 59, 1827.Google Scholar
Marshall, J., Adcroft, A., Hill, C., Perelman, L. & Heisey, C. 1997 A finite-volume, incompressible Navier–Stokes model for studies of the ocean on parallel computers. J. Geophys. Res 102 (C3), 57535766.Google Scholar
Moum, J. N., Nash, J. D. & Klymak, J. M. 2008 Small-scale processes in the coastal ocean. Oceanography 21, 2233.Google Scholar
Shroyer, E. L., Moum, J. N. & Nash, J. D. 2010 Mode 2 waves on the continental shelf: Ephemeral components of the nonlinear internal wavefield. J. Geophys. Res. 115, C07001.Google Scholar
Stastna, M. & Peltier, W. R. 2005 On the resonant generation of large-amplitude internal solitary and solitary-like waves. J. Fluid Mech. 543, 267292.CrossRefGoogle Scholar
Terletska, K., Jung, K. T., Talipova, T., Maderich, V., Brovchenko, I. & Grimshaw, R. H. J. 2016 Internal breather-like wave generation by the second mode solitary wave interaction with a step. Phys. Fluids 28, 116602.Google Scholar
Vlasenko, V., Stashchuk, N., Guo, C. & Chen, X. 2010 Multimodal structure of baroclinic tides in the south China sea. Nonlinear Process. Geophys. 17 (5), 529543.Google Scholar
Weideman, J. & Reddy, S. 2000 A matlab differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465519.Google Scholar
Yang, Y. J., Fang, Y. C., Chang, M.-H., Ramp, S. R., Kao, C.-C. & Tang, T. Y. 2009 Observations of second baroclinic mode internal solitary waves on the continental slope of the northern south china sea. J. Geophys. Res. 114, C10003.Google Scholar
Yang, Y. J., Fang, Y. C., Tang, T. Y. & Ramp, S. R. 2010 Convex and concave types of second baroclinic mode internal solitary waves. Nonlinear Process. Geophys. 17 (6), 605614.Google Scholar
Yih, C.-S. 1960 Gravity waves in a stratified fluid. J. Fluid Mech. 8 (04), 481508.Google Scholar
Zhou, X. & Grimshaw, R. 1989 The effect of variable currents on internal solitary waves. Dyn. Atmos. Oceans 14, 1739.Google Scholar