Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T12:18:36.408Z Has data issue: false hasContentIssue false
  • English
  • Français

Topographic effect on oblique internal wave–wave interactions

Published online by Cambridge University Press:  28 September 2018

C. Yuan Open the ORCID record for C. Yuan [Opens in a new window] ,
R. Grimshaw Open the ORCID record for R. Grimshaw [Opens in a new window] ,
E. Johnson Open the ORCID record for E. Johnson [Opens in a new window]  and
Z. Wang Open the ORCID record for Z. Wang [Opens in a new window]
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
Z. Wang
Affiliation:
Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, PR China
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Based on a variable-coefficient Kadomtsev–Petviashvili (KP) equation, the topographic effect on the wave interactions between two oblique internal solitary waves is investigated. In the absence of rotation and background shear, the model set-up featuring idealised shoaling topography and continuous stratification is motivated by the large expanse of continental shelf in the South China Sea. When the bottom is flat, the evolution of an initial wave consisting of two branches of internal solitary waves can be categorised into six patterns depending on the respective amplitudes and the oblique angles measured counterclockwise from the transverse axis. Using theoretical multi-soliton solutions of the constant-coefficient KP equation, we select three observed patterns and examine each of them in detail both analytically and numerically. The effect of shoaling topography leads to a complicated structure of the leading waves and the emergence of two types of trailing wave trains. Further, the case when the along-crest width is short compared with the transverse domain of interest is examined and it is found that although the topographic effect can still modulate the wave field, the spreading effect in the transverse direction is dominant.

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 856 , 10 December 2018 , pp. 36 - 60
Check for updates
Copyright
© 2018 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

Alford, M. H., MacKinnon, J. A., Nash, J. D., Simmons, H., Pickering, A., Klymak, J. M., Pinkel, R., Sun, O., Rainville, L., Musgrave, R., Beitzel, T., Fu, K.-H. & Lu, C.-W. 2011 Energy flux and dissipation in Luzon Strait: two tales of two ridges. J. Phys. Oceanogr. 41, 2211–2222.Google Scholar
Bogucki, D., Dickey, T. & Redekopp, L. G. 1997 Sediment resuspension and mixing by resonantly generated internal solitary waves. J. Phys. Oceanogr. 27, 1181–1196.Google Scholar
Cai, S. & Xie, J. 2010 A propagation model for the internal solitary waves in the northern South China Sea. J. Geophys. Res. 115 (C12), C12074.Google Scholar
Chakravarty, S. & Kodama, Y. 2008 Classification of the line-soliton solutions of KPII. J. Phys. A 41, 275209.Google Scholar
Chakravarty, S. & Kodama, Y. 2009 Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Maths 123, 83–151.Google Scholar
Chakravarty, S. & Kodama, Y. 2013 Construction of KP solitons from wave patterns. J. Phys. A 47, 025201.Google Scholar
Chen, G.-Y., Liu, C.-T., Wang, Y.-H. & Hsu, M.-K. 2011 Interaction and generation of long-crested internal solitary waves in the South China Sea. J. Geophys. Res. 116 (C6), C06013.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, 076601.Google Scholar
Duda, T. F., Lynch, J. F., Irish, J. D., Beardsley, R. C., Ramp, S. R., Chiu, C.-S., Tang, T. Y. & Yang, Y.-J. 2004 Internal tide and nonlinear internal wave behavior at the continental slope in the northern South China Sea. IEEE J. Ocean. Engng 29, 1105–1130.Google Scholar
Fu, K.-H., Wang, Y.-H., Laurent, L. S., Simmons, H. & Wang, D.-P. 2012 Shoaling of large-amplitude nonlinear internal waves at Dongsha Atoll in the northern South China Sea. Cont. Shelf Res. 37, 1–7.Google Scholar
Funakoshi, M. 1980 Reflection of obliquely incident solitary waves. J. Phys. Soc. Japan 49, 2371–2379.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., Pelinovsky, E. & Talipova, T. 2007 Modelling internal solitary waves in the coastal ocean. Surv. Geophys. 28, 273–298.Google Scholar
Grimshaw, R., Pelinovsky, E., Talipova, T. & Kurkina, O. 2010 Internal solitary waves: propagation, deformation and disintegration. Nonlinear Process. Geophys. 17, 633–649.Google Scholar
Guo, C. & Chen, X. 2014 A review of internal solitary wave dynamics in the northern South China Sea. Prog. Oceanogr. 121, 7–23.Google Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38 (1), 395–425.Google Scholar
Hsu, M.-K., Liu, A. K. & Liu, C. 2000 A study of internal waves in the China Seas and Yellow Sea using SAR. Cont. Shelf Res. 20, 389–410.Google Scholar
Jackson, C. R. 2004 An Atlas of Internal Solitary-like Waves and their Properties, 2nd edn. Global Ocean Associates. (Available at http://www.internalwaveatlas.com).Google Scholar
Jan, S., Lien, R.-C. & Ting, C.-H. 2008 Numerical study of baroclinic tides in Luzon Strait. J. Oceanogr. 64, 789–802.Google Scholar
Johnson, R. S. 1982 On the oblique interaction of a large and a small solitary wave. J. Fluid Mech. 120, 49–70.Google Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kao, C.-Y. & Kodama, Y. 2012 Numerical study of the KP equation for non-periodic waves. Math. Comput. Simul. 82, 1185–1218.Google Scholar
Klymak, J. M., Pinkel, R., Liu, C., Liu, A. K. & David, L. 2006 Prototypical solitons in the South China Sea. Geophys. Res. Lett. 33, L11607.Google Scholar
Kodama, Y. 2010 KP solitons in shallow water. J. Phys. A 43, 434004.Google Scholar
Kodama, Y., Oikawa, M. & Tsuji, H. 2009 Soliton solutions of the KP equation with V-shape initial waves. J. Phys. A 42, 312001.Google Scholar
Kodama, Y. & Yeh, H. 2016 The KP theory and Mach reflection. J. Fluid Mech. 800, 766–786.Google Scholar
Lamb, K. G. & Yan, L. 1996 The evolution of internal wave undular bores: comparisons of a fully nonlinear numerical model with weakly nonlinear theory. J. Phys. Oceanogr. 26, 2712–2734.Google Scholar
Li, W., Yeh, H. & Kodama, Y. 2011 On the Mach reflection and KP solitons in shallow water. J. Fluid Mech. 672, 326–357.Google Scholar
Liu, A. K. & Hsu, M.-k. 2004 Internal wave study in the South China Sea using synthetic aperture radar (SAR). Intl J. Remote Sens. 25, 1261–1264.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
Melville, W. K. 1980 On the Mach reflexion of a solitary wave. J. Fluid Mech. 98, 285–297.Google Scholar
Miles, J. W. 1977a Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–169.Google Scholar
Miles, J. W. 1977b Resonantly interacting solitary waves. J. Fluid Mech. 79, 171–179.Google Scholar
Oikawa, M., Tsuji, H., Kodama, Y. & Maruno, K. 2010 Soliton solution of KPII equation and its application. In Integrable Systems and their Applications (ed. Isojima, Shin), pp. 65–84. RIMS Kokyuroku (in Japanese).Google Scholar
Osborne, A., Burch, T. & Scarlet, R. 1978 The influence of internal waves on deep-water drilling. J. Petrol. Tech. 30, 1497–1504.Google Scholar
Pierini, S. 1989 A model for the Alboran Sea internal solitary waves. J. Phys. Oceanogr. 19, 755–772.Google Scholar
Ramp, S. R., Tang, T. Y., Duda, T. F., Lynch, J. F., Liu, A. K., Chiu, C.-S., Bahr, F. L., Kim, H.-R. & Yang, Y.-J. 2004 Internal solitons in the northeastern South China Sea. Part I. Sources and deep water propagation. IEEE J. Ocean. Engng 29, 1157–1181.Google Scholar
Schlatter, P., Adams, N. A. & Kleiser, L. 2005 A windowing method for periodic inflow/outflow boundary treatment of non-periodic flows. J. Comput. Phys. 206, 505–535.Google Scholar
Shimizu, K. & Nakayama, K. 2017 Effects of topography and Earth’s rotation on the oblique interaction of internal solitary-like waves in the Andaman Sea. J. Geophys. Res. 122, 7449–7465.Google Scholar
Tanaka, M. 1993 Mach reflection of a large-amplitude solitary wave. J. Fluid Mech. 248, 637–661.Google Scholar
Tsuji, H. & Oikawa, M. 2007 Oblique interaction of solitons in an extended Kadomtsev–Petviashvili equation. J. Phys. Soc. Japan 76, 084401.Google Scholar
Wang, C. & Pawlowicz, R. 2012 Oblique wave-wave interactions of nonlinear near-surface internal waves in the Strait of Georgia. J. Geophys. Res. 117 (C6), C06031.Google Scholar
Wang, Y.-H., Dai, C.-F. & Chen, Y.-Y. 2007 Physical and ecological processes of internal waves on an isolated reef ecosystem in the South China Sea. Geophys. Res. Lett. 34 (18), L18609.Google Scholar
Xie, X., Li, M., Scully, M. & Boicourt, W. C. 2017 Generation of internal solitary waves by lateral circulation in a stratified estuary. J. Phys. Oceanogr. 47, 1789–1797.Google Scholar
Xue, J., Graber, H. C., Romeiser, R. & Lund, B. 2014 Understanding internal wave–wave interaction patterns observed in satellite images of the Mid-Atlantic Bight. IEEE Trans. Geosci. Remote Sens. 52, 3211–3219.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, 1182–1199.Google Scholar
Yeh, H. & Li, W. 2014 Laboratory realization of KP-solitons. J. Phys.: Conf. Ser. 482, 012046.Google Scholar
Yeh, H., Li, W. & Kodama, Y. 2010 Mach reflection and KP solitons in shallow water. Eur. Phys. J. 185, 97–111.Google Scholar
Yuan, C., Grimshaw, R. & Johnson, E. 2018a The evolution of second mode internal solitary waves over variable topography. J. Fluid Mech. 836, 238–259.Google Scholar
Yuan, C., Grimshaw, R., Johnson, E. & Chen, X. 2018b The propagation of internal solitary waves over variable topography in a horizontally two-dimensional framework. J. Phys. Oceanogr. 48, 283–300.Google Scholar
Zhao, Z., Klemas, V., Zheng, Q. & Yan, X.-H. 2004 Remote sensing evidence for baroclinic tide origin of internal solitary waves in the northeastern South China Sea. Geophys. Res. Lett. 31, L06302.Google Scholar
Zheng, Q., Klemas, V., Yan, X.-H., Wang, Z. & Kagleder, K. 1997 Digital orthorectification of space shuttle coastal ocean photographs. Intl J. Remote Sens. 18, 197–211.Google Scholar