Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T01:59:55.996Z Has data issue: false hasContentIssue false
  • English
  • Français

Generation of mode 2 internal waves by the interaction of mode 1 waves with topography

Published online by Cambridge University Press:  18 October 2019

Zihua Liu Open the ORCID record for Zihua Liu [Opens in a new window] ,
Roger Grimshaw Open the ORCID record for Roger Grimshaw [Opens in a new window]  and
Edward Johnson Open the ORCID record for Edward Johnson [Opens in a new window]
Zihua Liu*
Affiliation:
Department of Mathematics, University College London, London, WC1E 6BT, UK
Roger Grimshaw
Affiliation:
Department of Mathematics, University College London, London, WC1E 6BT, UK
Edward Johnson
Affiliation:
Department of Mathematics, University College London, London, WC1E 6BT, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Oceanic internal waves can be decomposed into an infinite set of modes, and the dominant internal mode 1 waves have been extensively investigated. Although mode 2 waves have been observed, they have not received comparable attention, especially the generation mechanisms. In this work, we examine the generation of mode 2 internal waves by the interaction of mode 1 waves with topography. We use a coupled linear long-wave theory with mode coupling through topography, combined with evolution using a Korteweg–de Vries model, to predict the mode 2 wave amplitude, in an ideal three-layer fluid model, in a smooth density stratification and in two realistic oceanic settings. We find that the mode 2 wave amplitude is usually much smaller than the incident mode 1 wave amplitude and is quite sensitive to the pycnocline thickness, topographic slope and background stratification.

JFM classification

Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Solitary waves Geophysical and Geological Flows: Topographic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 880 , 10 December 2019 , pp. 799 - 830
Copyright
© 2019 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, 279–298.Google Scholar
Akylas, T. R., Grimshaw, R. H. J., Clarke, S. R. & Tabaei, A. 2007 Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline. J. Fluid Mech. 593, 297–313.Google Scholar
Deepwell, D., Stastna, M., Carr, M. & Davies, P. A. 2017 Interaction of a mode-2 internal solitary wave with narrow isolated topography. Phys. Fluids 29 (7), 076601.Google Scholar
Farmer, D. M. & Smith, J. D. 1980 Tidal interaction of stratified flow with a sill in Knight Inlet. Deep-Sea Res. A 27 (3), 239–254.Google Scholar
Gerkema, T. 2001 Internal and interfacial tides: beam scattering and local generation of solitary waves. J. Mar. Res. 59 (2), 227–255.Google Scholar
Griffiths, S. D. & Grimshaw, R. H. J. 2007 Internal tide generation at the continental shelf modeled using a modal decomposition: two-dimensional results. J. Phys. Ocean. 37, 428–451.Google Scholar
Grimshaw, R. 1981 Evolution equations for long nonlinear internal waves in stratified shear flows. Stud. Appl. Maths 65, 159–188.Google Scholar
Grimshaw, R. 2007 Internal solitary waves in a variable medium. Gesellschaft fur Angewandte Mathematik 30, 96–109.Google Scholar
Grimshaw, R. & Helfrich, K. R. 2018 Internal solitary wave generation by tidal flow over topography. J. Fluid Mech. 839, 387–407.Google Scholar
Grimshaw, R., Pelinovsky, E., Talipova, T. & Kurkina, A. 2010 Internal solitary waves: propagation, deformation and disintegration. Nonlinear Process. Geophys. 17, 633–649.Google Scholar
Helfrich, K. R. & Melville, W. K. 1986 On long nonlinear internal waves over slope-shelf topography. J. Fluid Mech. 167, 285–308.Google Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.Google Scholar
Helland-Hansen, B. & Nansen, F. 1909 The Norwegian Sea — its Physical Oceanography based upon the Norwegian Researches 1900–1904. (Report on Norwegian Fishery and Marine Investigations, vol. II, No. 2). Det Mallingske Bogtrykkeri.Google Scholar
Klymak, J. M. & Moum, J. N. 2003 Internal solitary waves of elevation advancing on a shoaling shelf. Geophys. Res. Lett. 30 (20), doi:10.1029/2003GL017706.Google Scholar
Lamb, K. G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231–254.Google Scholar
Lamb, K. G. & Warn-Varnas, A. 2015 Two-dimensional numerical simulations of shoaling internal solitary waves at the ASIAEX site in the South China Sea. Nonlinear Process. Geophys. 22 (3), 289–312.Google Scholar
Liang, J., Du, T., Li, X. & He, M. 2018 Generation of mode-2 internal waves in a two-dimensional stratification by a mode-1 internal wave. Wave Motion 83, 227–240.Google Scholar
Liang, J. & Li, X. 2019 Generation of second-mode internal solitary waves during winter in the northern South China Sea. Ocean Dyn. 69 (3), 313–321.Google Scholar
Liu, A. K., Chang, Y. S., Hsu, M.-K. & Liang, N. K. 1998 Evolution of nonlinear internal waves in the East and South China Seas. J. Geophys. Res. 103 (C4), 7995–8008.Google Scholar
Liu, Z., Grimshaw, R. & Johnson, E. 2019 The interaction of a mode-1 internal solitary wave with a step and the generation of mode-2 waves. Geophys. Astrophys. Fluid Dyn. 113 (4), 327–347.Google 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, 18–27.Google Scholar
Marshall, J., Adcroft, A., Hill, C., Perelman, L. & Heisey, C. 1997a A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res. 102 (C3), 5753–5766.Google Scholar
Marshall, J., Hill, C., Perelman, L. & Adcroft, A. 1997b Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res. 102 (C3), 5733–5752.Google Scholar
Maxworthy, T. 1980 On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions. J. Fluid Mech. 96 (1), 47–64.Google Scholar
Mehta, A. P., Sutherland, B. R. & Kyba, P. J. 2002 Interfacial gravity currents. II. Wave excitation. Phys. Fluids 14 (10), 3558–3569.Google Scholar
Moum, J. N. & Smyth, W. D. 2006 The pressure disturbance of a nonlinear internal wave train. J. Fluid Mech. 558, 153–177.Google Scholar
Ramp, S. R., Yang, Y. J., Reeder, D. B. & Bahr, F. L. 2012 Observations of a mode-2 nonlinear internal wave on the northern Heng-Chun Ridge south of Taiwan. J. Geophys. Res. 117 (C3), doi:10.1029/2011JC007662.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 wave field. J. Geophys. Res. 115 (C7), doi:10.1029/2009JC005605.Google Scholar
Shroyer, E. L., Moum, J. N. & Nash, J. D. 2011 Nonlinear internal waves over New Jersey’s continental shelf. J. Geophys. Res. 116 (C3), doi:10.1029/2010JC006332.Google Scholar
Terletska, K., Jung, K. T., Talipova, T., Maderich, V., Brovchenko, I. & Grimshaw, R. 2016 Internal breather-like wave generation by the second mode solitary wave interaction with a step. Phys. Fluids 28, 116602.Google Scholar
Vlasenko, V. I. & Hutter, K. 2001 Generation of second mode solitary waves by the interaction of a first mode soliton with a sill. Nonlinear Process. Geophys. 8, 223–239.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 (C10), doi:10.1029/2009JC005318.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), 605–614.Google Scholar
Yang, Y.-J., Tang, T. Y., Chang, M. H., Liu, A. K., Hsu, M.-K. & Ramp, S. R. 2004 Solitons northeast of Tung-Sha Island during the ASIAEX pilot studies. IEEE J. Ocean. Engng 29 (4), 1182–1199.Google Scholar
Yuan, C., Grimshaw, R. & Johnson, E. 2018 The propagation of second mode internal solitary waves over variable topography. J. Fluid Mech. 836, 238–259.Google Scholar
Zhao, Z. & Alford, M. H. 2006 Source and propagation of internal solitary waves in the northeastern South China Sea. J. Geophys. Res. 111 (C11), doi:10.1029/2006JC003644.Google Scholar