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The surface signature of internal waves

Published online by Cambridge University Press:  31 August 2012

W. Craig ,
P. Guyenne  and
C. Sulem
W. Craig*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada
P. Guyenne
Affiliation:
Department of Mathematical Sciences, University of Delaware, DE 19716, USA
C. Sulem
Affiliation:
Department of Mathematics, University of Toronto, ON, M5S 3G3, Canada
*
†Email address for correspondence: craig@math.mcmaster.ca
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Abstract

Oceans that are stratified by density into distinct layers support internal waves. An internal soliton gives rise to characteristic features on the surface, a signature of its presence, in the form of a ‘rip’ region, as reported in Osborne & Burch (Science, vol. 208, 1980, pp. 451–460), which results in a change in reflectance as seen in NASA photographs from the space shuttle. In the present paper, we give a new analysis of this signature of an internal soliton, and the ‘mill pond’ effect of an almost completely calm sea after its passage. Our analysis models the resonant interaction of nonlinear internal waves with the surface modes, where the surface signature is generated by a process analogous to radiative absorption. These theoretical results are illustrated with numerical simulations that take oceanic parameters into account.

JFM classification

Mathematical Foundations: Hamiltonian theory Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 710 , 10 November 2012 , pp. 277 - 303
Copyright
Copyright © Cambridge University Press 2012

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References

1. Alpers, W. 1985 Theory of radar imaging of internal waves. Nature 314, 245–247.CrossRefGoogle Scholar
2. Bakhanov, V. V. & Ostrovsky, L. A. 2002 Action of strong internal solitary waves on surface waves. J. Geophys. Res. 107, 3139.Google Scholar
3. Basovich, A. Ya. & Bahanov, V. V. 1984 Surface wave kinematics in the field of an internal wave. Izv. Atmos. Ocean Phys. 20, 50–54.Google Scholar
4. Basovich, A. Ya. & Talanov, V. I. 1977 Transformation of short surface waves on inhomogeneous currents. Izv. Atmos. Ocean Phys. 13, 514–519.Google Scholar
5. Benjamin, T. B. & Bridges, T. J. 1997 Reappraisal of the Kelvin–Helmholtz problem. I. Hamiltonian structure. J. Fluid Mech. 333, 301–325.CrossRefGoogle Scholar
6. Bourgault, D. & Kelley, D. E. 2003 Wave-induced boundary mixing in a partially mixed estuary. J. Mar. Res. 61, 553–576.CrossRefGoogle Scholar
7. 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. J. Geophys. Res. 93, 12249–12263.CrossRefGoogle Scholar
8. Chadan, K., Colton, D., Paivarinta, L. & Rundell, W. 1997 An Introduction to Inverse Scattering and Inverse Spectral Problems. SIAM.CrossRefGoogle Scholar
9. Craig, W., Guyenne, P. & Kalisch, H. 2005 Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Maths 58, 1587–1641.CrossRefGoogle Scholar
10. Craig, W., Guyenne, P. & Sulem, C. 2010 A Hamiltonian approach to nonlinear modulation of surface water waves. Wave Motion 47, 552–563.CrossRefGoogle Scholar
11. Craig, W., Guyenne, P. & Sulem, C. 2011 Coupling between internal and surface waves. Nat. Hazards 57, 617–642.CrossRefGoogle Scholar
12. Craig, W., Schanz, U. & Sulem, C. 1997 The modulational regime of three-dimensional water waves and the Davey–Stewartson system. Ann. Inst. Henri Poincaré (C) Nonlin. Anal. 14, 615–667.Google Scholar
13. Djordjevic, V. D. & Redekopp, R. G. 1977 On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79, 703–714.CrossRefGoogle Scholar
14. Donato, A. N., Peregrine, D. H. & Stocker, J. R. 1999 The focusing of surface waves by internal waves. J. Fluid Mech. 384, 27–58.CrossRefGoogle Scholar
15. Evans, W. A. B. & Ford, M. J. 1996 An integral equation approach to internal (2-layer) solitary waves. Phys. Fluids 8, 2032–2047.CrossRefGoogle Scholar
16. 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, 1982–1995.CrossRefGoogle Scholar
17. Gargett, A. E. & Hughes, B. A. 1972 On the interaction of surface and internal waves. J. Fluid Mech. 52, 179–191.CrossRefGoogle Scholar
18. Gasparovic, R. F., Apel, J. R. & Kasischke, E. S. 1988 An overview of the SAR internal wave signature experiment. J. Geophys. Res. 93, 12304–12316.CrossRefGoogle Scholar
19. Gear, J. & Grimshaw, R. 1984 Weak and strong interactions between internal solitary waves. Stud. Appl. Maths 70, 235–258.CrossRefGoogle Scholar
20. Hashizume, Y. 1980 Interaction between short surface waves and long internal waves. J. Phys. Soc. Japan 48, 631–638.CrossRefGoogle Scholar
21. Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
22. Kawahara, T., Sugimoto, N. & Kakutani, T. 1975 Nonlinear interaction between short and long capillary-gravity waves. J. Phys. Soc. Japan 39, 1379–1386.CrossRefGoogle Scholar
23. Lee, K.-J., Hwung, H.-H., Yang, R.-Y. & Shugan, I. V. 2007 Stokes wave modulation by internal waves. Geophys. Res. Lett. 34, L23605.CrossRefGoogle Scholar
24. 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, 773–800.CrossRefGoogle Scholar
25. Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stress in water waves, a physical discussion with applications. Deep-Sea Res. 11, 529–562.Google Scholar
26. Ma, Y.-C. 1983 A study of resonant interactions between internal and surface waves based on a two-layer fluid model. Wave Motion 5, 145–155.CrossRefGoogle Scholar
27. Marston, C. C. & Balint-Kurti, G. G. 1989 The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions. J. Chem. Phys. 91, 3571–3576.CrossRefGoogle Scholar
28. Osborne, A. R. & Burch, T. L. 1980 Internal solitons in the Andaman Sea. Science 208, 451–460.CrossRefGoogle ScholarPubMed
29. Părău, E. & Dias, F. 2001 Interfacial periodic waves of permanent form with free-surface boundary conditions. J. Fluid Mech. 437, 325–336.CrossRefGoogle Scholar
30. Perry, B. R. & Schimke, G. R. 1965 Large-amplitude internal waves observed off the northwest coast of Sumatra. J. Geophys. Res. 70, 2319–2324.CrossRefGoogle Scholar
31. Peters, A. S. & Stoker, J. J. 1960 Solitary waves in liquids having non-constant density. Commun. Pure Appl. Maths 13, 115–164.CrossRefGoogle Scholar
32. Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar