Internal Solitary Waves in a Layered Weakly Stratified Flow

  • Nikolay Makarenko
  • Janna Maltseva
  • Roman Tarakanov
  • Kseniya Ivanova
Part of the Springer Oceanography book series (SPRINGEROCEAN)


The problem on internal waves in a weakly stratified two-layered flow is studied semi-analytically. The long-wave model describing travelling waves is constructed by means of scaling procedure with a small Boussinesq parameter. It is demonstrated that solitary wave regimes can be affected by the Kelvin–Helmholtz instability arising due to interfacial velocity shear in the upstream flow.



This work was supported by Russian Foundation for Basic Research (grant No 15-01-03942). RYuT acknowledges the support by Russian Sciences Foundation (grant No 16–17-10149).


  1. 1.
    Benney, D. J., & Ko, D. R. S. (1978). The propagation of long large amplitude internal waves. Studies in Applied Mathematics, 59, 187–199.CrossRefGoogle Scholar
  2. 2.
    Choi, W., & Camassa, R. (1999). Fully nonlinear internal waves in a two-fluid system. Journal of Fluid Mechanics, 396, 1–36.CrossRefGoogle Scholar
  3. 3.
    Drazin, P. (2002). Introduction to hydrodynamic stabilitry. Cambridge, UK: Cambridge University Press.Google Scholar
  4. 4.
    Gavrilyuk, S. L., Makarenko, N. I., & Sukhinin, S. V. (2017). Waves in continuous media. In: Lecture Notes in Geosystem Mathematics and Computing. Cham, Switzerland: Birkhäuser/Springer.Google Scholar
  5. 5.
    Grue, J., Jensen, A., Rusås, P. O., & Sveen, J. K. (2000). Breaking and broadening of internal solitary waves. Journal of Fluid Mechanics, 413, 181–217.CrossRefGoogle Scholar
  6. 6.
    Helfrich, K. R., & Melville, W. K. (2006). Long nonlinear internal waves. Annual Review of Fluid Mechanics, 38, 395–425.CrossRefGoogle Scholar
  7. 7.
    Kakutani, T., & Yamasaki, N. (1978). Solitary waves on a two-layer fluid. Journal of the Physical Society of Japan, 45, 674–679.CrossRefGoogle Scholar
  8. 8.
    Lamb, K., & Wan, B. (1998). Conjugate flows and flat solitary waves for a continuously stratified fluid. Physics of Fluids, 10, 2061–2079.CrossRefGoogle Scholar
  9. 9.
    LeBlond, P. H., & Mysak, L. A. (1978). Waves in the ocean. Amsterdam: Elsevier.Google Scholar
  10. 10.
    Long, R. R. (1965). On the Boussinesq approximation and its role in the theory of internal waves. Tellus, 17(1), 46–52.CrossRefGoogle Scholar
  11. 11.
    Makarenko, N. I., & Maltseva, J. L. (2008). An analytical model of large amplitude internal solitary waves. In E. Pelinovsky & C. Kharif (Eds.), Extreme Ocean Waves (pp. 179–189). Dordrecht: Springer.Google Scholar
  12. 12.
    Makarenko, N. I., & Maltseva, J. L. (2009a). Phase velocity spectrum of internal waves in a weakly-stratified two-layer fluid. Fluid Dynamics, 44(2), 278–294.CrossRefGoogle Scholar
  13. 13.
    Makarenko, N. I., & Maltseva, J. L. (2009b). Solitary waves in a weakly stratified two-layer fluid. Journal of Applied Mechanics and Technical Physics, 50(2), 229–234.CrossRefGoogle Scholar
  14. 14.
    Makarenko, N. I., Maltseva, J. L., & Kazakov, A. Y. (2009). Conjugate flows and amplitude bounds for inernal solitary waves. Nonlinear Processes in Geophysics, 16, 169–178.Google Scholar
  15. 15.
    Miyata, M. (1985). An internal solitary wave of large amplitude. La Mer, 23(2), 43–48.Google Scholar
  16. 16.
    Morozov, E. G. (1995). Semidiurnal internal wave global field. Deep Sea Research, 42(1), 135–148.CrossRefGoogle Scholar
  17. 17.
    Morozov, E., Demidov, A., Tarakanov, R., & Zenk, W. (2010). Abyssal channels in the Atlantic Ocean: Water structure and flows. Dordrecht: Springer.CrossRefGoogle Scholar
  18. 18.
    Morozov, E. G., Tarakanov, R Yu., Lyapidevskii, V Yu., & Makarenko, N. I. (2012). Abyssal cataracts in the Romanche and Chain fracture zones. Doklady Earth Sciences, 446, 1211–14.CrossRefGoogle Scholar
  19. 19.
    Ovsyannikov, L. V., Makarenko, N. I., Nalimov, V. I., et al. (1985). Nonlinear problems of the theory of surface and internal waves. Novosibirsk: Nauka (in Russian).Google Scholar
  20. 20.
    Pelinovsky, E. N., Polukhina, O., Slyunaev, A., & Talipova, T. (2007). Internal solitary waves. In R. Grimshaw (Ed.), Solitary waves in fluids (pp. 85–110). Southhampton: WIT Press.CrossRefGoogle Scholar
  21. 21.
    Thorpe, S. A. (1985). Laboratory observations of secondary structures in Kelvin–Helmholtz billows and consequences for ocean mixing. Geophysical & Astrophysical Fluid Dynamics, 34, 175–190.CrossRefGoogle Scholar
  22. 22.
    Turner, J. S. (1973). Buoyancy effects in fluid. Cambridge UK: Cambridge University Press.CrossRefGoogle Scholar
  23. 23.
    Turner, R. E. L., & VandenBroeck, J.-M. (1988). Broadening of interfacial solitary waves. Physics of Fluids, 31, 2486–2490.CrossRefGoogle Scholar
  24. 24.
    Van Haren, H., Gostiaux, L., Morozov, E., & Tarakanov, R. (2014). Extremely long Kelvin–Helmholtz billow trains in the Romanche Fracture zone. Geophysical Research Letters, 44(23), 8445–8451.CrossRefGoogle Scholar
  25. 25.
    Voronovich, A. G. (2003). Strong solitary internal waves in a 2.5- layer model. Journal of Fluid Mechanics, 474, 85–94.CrossRefGoogle Scholar
  26. 26.
    Yih, C. S. (1980). Stratified Flows. New York: Academic Press.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Nikolay Makarenko
    • 1
  • Janna Maltseva
    • 1
  • Roman Tarakanov
    • 2
  • Kseniya Ivanova
    • 3
  1. 1.Lavrentyev Institute of HydrodynamicsNovosibirsk State UniversityNovosibirskRussia
  2. 2.Shirshov Institute of OceanologyMoscowRussia
  3. 3.Aix-Marseille UniversiteMarseille 06France

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