Skip to main content Accessibility help
×
Home

Modulation-resonance mechanism for surface waves in a two-layer fluid system

  • Shixiao W. Jiang (a1), Gregor Kovačič (a2), Douglas Zhou (a3) and David Cai (a3) (a4) (a5)

Abstract

We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an underlying, slowly varying, long-wavelength baroclinic flow in a two-layer, density-stratified system. The results of our model show numerically that, under strong nonlinearity, surface waves, with their typical wavenumber being the resonant $k_{res}$ , can be generated locally at the leading edge of the underlying, slowly varying, long-wavelength baroclinic flow. Here, the resonant $k_{res}$ satisfies the class 3 triad resonance condition among two short-mode waves and one long-mode wave in which all waves propagate in the same direction. Moreover, when the slope of the baroclinic flow is sufficiently small, only one spatially localized large-amplitude surface wave packet can be generated at the leading edge. This localized surface wave packet becomes high in amplitude and large in group velocity after the interaction with its surrounding waves. These results are qualitatively consistent with various experimental observations including resonant surface waves at the leading edge of an internal wave. Subsequently, we propose a mechanism, referred to as the modulation-resonance mechanism, underlying these surface phenomena, based on our numerical simulations. The proposed modulation-resonance mechanism combines the linear modulation, ray-based, theory for the spatiotemporal asymmetric behaviour of surface waves and the nonlinear class 3 triad resonance theory for the energy focusing of surface waves around the resonant wavenumber $k_{res}$ in Fourier space.

Copyright

Corresponding author

Email addresses for correspondence: suj235@psu.edu, kovacg@rpi.edu, zdz@sjtu.edu.cn

References

Hide All
Alam, M.-R. 2012 A new triad resonance between co-propagating surface and interfacial waves. J. Fluid Mech. 691, 267278.10.1017/jfm.2011.473
Alford, M. H., Peacock, T., MacKinnon, J. A., Nash, J. D., Buijsman, M. C., Centuroni, L. R., Chao, S.-Y., Chang, M.-H., Farmer, D. M., Fringer, O. B. et al. 2015 The formation and fate of internal waves in the South China sea. Nature 521 (7550), 6569.10.1038/nature14399
Alpers, W. 1985 Theory of radar imaging of internal waves. Nature 314 (6008), 245247.10.1038/314245a0
Apel, J. R., Ostrovsky, L. A., Stepanyants, Y. A. & Lynch, J. F. 2007 Internal solitons in the ocean and their effect on underwater sound. J. Acoust. Soc. Am. 121, 695722.10.1121/1.2395914
Bakhanov, V. V. & Ostrovsky, L. A. 2002 Action of strong internal solitary waves on surface waves. J. Geophys. Res. 107, 3139.10.1029/2001JC001052
Barros, R. & Choi, W. 2009 Inhibiting shear instability induced by large amplitude internal solitary waves in two-layer flows with a free surface. Stud. Appl. Maths 122, 325346.10.1111/j.1467-9590.2009.00436.x
Barros, R. & Gavrilyuk, S. 2007 Dispersive nonlinear waves in two-layer flows with free surface part ii. large amplitude solitary waves embedded into the continuous spectrum. Stud. Appl. Maths 119, 213251.10.1111/j.1467-9590.2007.00384.x
Benney, D. 1977 A general theory for interactions between short and long waves. Stud. Appl. Maths 56 (1), 8194.10.1002/sapm197756181
Caponi, E. A., Crawford, D. R., Yuen, H. C. & Saffman, P. G.1988 Modulation of radar backscatter from the ocean by a variable surface current. Tech. Rep. DTIC Document.10.1029/JC093iC10p12249
Chen, T.2005 An efficient algorithm based on quadratic spline collocation and finite difference methods for parabolic partial differential equations. PhD thesis, University of Toronto.
Choi, W., Barros, R. & Jo, T.-C. 2009 A regularized model for strongly nonlinear internal solitary waves. J. Fluid Mech. 629, 7385.10.1017/S0022112009006594
Choi, W. & Camassa, R. 1996 Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313, 83103.10.1017/S0022112096002133
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 136.10.1017/S0022112099005820
Craig, W., Guyenne, P. & Kalisch, H. 2004 A new model for large amplitude long internal waves. C. R. Méc 332, 525530.10.1016/j.crme.2004.02.026
Craig, W., Guyenne, P. & Kalisch, H. 2005 Hamiltonian long wave expansions for free surfaces and interfaces. Commun. Pure Appl. Maths 58, 15871641.10.1002/cpa.20098
Craig, W., Guyenne, P. & Sulem, C. 2011 Coupling between internal and surface waves. Nat. Hazards 57, 617642.10.1007/s11069-010-9535-4
Craig, W., Guyenne, P. & Sulem, C. 2012 The surface signature of internal waves. J. Fluid Mech. 710, 277303.10.1017/jfm.2012.364
Donato, A. N., Peregrine, D. H. & Stocker, J. R. 1999 The focusing of surface waves by internal waves. J. Fluid Mech. 384, 2758.10.1017/S0022112098003917
Duda, T. F. & Farmer, D. M.1999 The 1998 WHOI/IOS/ONR Internal Solitary Wave Workshop: Contributed Papers. Tech. Rep. DTIC Document.10.21236/ADA368664
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 wave behavior in the continental slope in the northern South China sea. IEEE J. Ocean. Engng 29, 11051131.10.1109/JOE.2004.836998
Dysthe, K., Krogstad, H. E. & Müller, P. 2008 Oceanic rogue waves. Annu. Rev. Fluid Mech. 40, 287310.10.1146/annurev.fluid.40.111406.102203
Funakoshi, M. & Oikawa, M. 1983 The resonant interaction between a long internal gravity wave and a surface gravity wave packet. J. Phys. Soc. Japan 56, 19821995.10.1143/JPSJ.52.1982
Gargett, A. E. & Hughes, B. A. 1972 On the interaction of surface and internal waves. J. Fluid Mech. 52 (01), 179191.10.1017/S0022112072003027
Gasparovic, R. F., Apel, J. R. & Kasischke, E. S. 1988 An overview of the SAR internal wave experiment. J. Geophys. Res. 93, 1230412316.10.1029/JC093iC10p12304
Guyenne, P. 2006 Large-amplitude internal solitary waves in a two-fluid model. C. R. Méc. 334 (6), 341346.
Han, H. & Xu, Z. 2007 Numerical solitons of generalized Korteweg–de Vries equations. Appl. Maths Comput. 186, 483489.10.1016/j.amc.2006.07.111
Hashizume, Y. 1980 Interaction between short surface waves and long internal waves. J. Phys. Soc. Japan 48, 631638.10.1143/JPSJ.48.631
Hwung, H.-H., Yang, R.-Y. & Shugan, I. V. 2009 Exposure of internal waves on the sea surface. J. Fluid Mech. 626, 120.10.1017/S0022112008004758
Jo, T.-C. & Choi, W. 2008 On stabilizing the strongly nonlinear internal wave model. Stud. Appl. Maths 120, 6585.10.1111/j.1467-9590.2007.00393.x
Johnston, T. S., Merrifield, M. A. & Holloway, P. E. 2003 Internal tide scattering at the line islands ridge. J. Geophys. Res. 108 (C11), 3365.10.1029/2003JC001844
Kawahara, T., Sugimoto, N. & Kakutani, T. 1975 Nonlinear interaction between short and long capillary-gravity waves. J. Phys. Soc. Japan 39 (5), 13791386.10.1143/JPSJ.39.1379
Kodaira, T., Waseda, T., Miyata, M. & Choi, W. 2016 Internal solitary waves in a two-fluid system with a free surface. J. Fluid Mech. 804, 201223.10.1017/jfm.2016.510
Koop, C. G. & Butler, G. 1981 An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112, 225251.10.1017/S0022112081000372
Kropfli, R. A., Ostrovski, L. A., Stanton, T. P., Skirta, E. A., Keane, A. N. & Irisov, V. 1999 Relationships between strong internal waves in the coastal zone and their radar and radiometric signatures. J. Geophys. Res. 104, 31333148.10.1029/98JC02549
Lee, K.-J., Shugan, I. V. & An, J.-S. 2007 On the interaction between surface and internal waves. J. Korean Phys. Soc. 51, 616622.10.3938/jkps.51.616
Lewis, J. E., Lake, B. M. & Ko, D. R. S. 1974 On the interaction of internal waves and surface gravity waves. J. Fluid Mech. 63 (04), 773800.10.1017/S0022112074002199
Moore, S. E. & Lien, R.-C. 2007 Pilot whales follow internal solitary waves in the South China sea. Mar. Mam. Sci. 23 (1), 193196.10.1111/j.1748-7692.2006.00086.x
Müller, P., Garrett, C. & Osborne, A. 2005 Rogue waves. Oceanography 18 (3), 66.10.5670/oceanog.2005.30
Osborne, A. R. & Burch, T. L. 1980 Internal solitons in the andaman sea. Science 208, 451460.10.1126/science.208.4443.451
Parau, E. & Dias, F. 2001 Interfacial periodic waves of permanent form with free-surface boundary conditions. J. Fluid Mech. 437, 325336.10.1017/S0022112001004220
Perry, R. B. & Schimke, G. R. 1965 Large-amplitude internal waves observed off the northwest coast of sumatra. J. Geophys. Res. 70 (10), 23192324.10.1029/JZ070i010p02319
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Phillips, O. M. 1974 Nonlinear dispersive waves. Annu. Rev. Fluid Mech. 6, 93110.10.1146/annurev.fl.06.010174.000521
Sepulveda, N. 1987 Solitary waves in the resonant phenomenon between a surface gravity wave packet and an internal gravity wave. Phys. Fluids 30 (7), 19841992.10.1063/1.866212
Tanaka, M. & Wakayama, K. 2015 A numerical study on the energy transfer from surface waves to interfacial waves in a two-layer fluid system. J. Fluid Mech. 763, 202217.10.1017/jfm.2014.668
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed