Skip to main content
SpringerLink
Log in

Dynamics of Nonstationary Cylindrical Solitary Internal Waves

  • Published:
Izvestiya, Atmospheric and Oceanic Physics Aims and scope Submit manuscript

Abstract

An approximate analytical description of the nonstationary evolution of cylindrical nonlinear solitary waves with a complex structure is given. A modified Gardner equation with a boundary condition in the form of a “wide” soliton close to the limiting one is analyzed. The analysis shows a qualitative difference in the behavior of converging and diverging waves, as well as a difference from the quasi-stationary dynamics of cylindrical solitons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.

Similar content being viewed by others

REFERENCES

  1. C. R. Jason, J. C. DaSilva, G. A. Jeans, W. Alpers, and M. J. Caruso, “Nonlinear internal waves in synthetic aperture radar imagery,” Oceanography 26, 68–79 (2013).

    Article  Google Scholar 

  2. R. A. Kropfli, L. A. Ostrovsky, T. P. Stanton, E. A. Skirta, A. N. Keane, and V. Irisov, “Relationships between strong internal waves in the coastal zone and their radar and radiometric signatures,” J. Geophys. Res. 104 (C2), 3133–3148 (1999).

    Article  Google Scholar 

  3. D. M. Farmer and L. Armi, “The flow of Atlantic water through the Strait of Gibraltar,” Prog. Oceanogr. 21, 1–105 (1988).

    Article  Google Scholar 

  4. S. Maxon and J. Viecelli, “Cylindrical solitons,” Phys. Fluids 17, 1614–1616 (1974).

    Article  Google Scholar 

  5. R. S. Johnson, “Water waves and Korteweg–de Vries equations,” J. Fluid Mech. 97, 701–719 (1980).

    Article  Google Scholar 

  6. A. A. Dorfman, E. N. Pelinovskii, and Yu. A. Stepanyants, “Finite-amplitude cylindrical and spherical waves in weakly dispersive media,” Sov. Phys. J. Appl. Mech. Tech. Phys., No. 2, 206–211 (1981).

  7. Yu. A. Stepanyants, “Experimental investigation of cylindrically diverging solitons in an electric lattice,” Wave Motion, No. 3, 335–341 (1981).

    Article  Google Scholar 

  8. P. D. Weidman and R. Zakhem, “Cylindrical solitary waves,” J. Fluid Mech. 191, 557–573 (1988).

    Article  Google Scholar 

  9. R. S. Johnson, “Ring waves on the surface of shear flows: A linear and nonlinear theory,” J. Fluid Mech. 215, 145–160 (1990).

    Article  Google Scholar 

  10. Yu. A. Stepanyants, “On the attenuation of internal solitary waves due to cylindrical divergence,” Izv. Akad. Nauk SSSR: Fiz. Atmos. Okeana 17 (8), 886–888 (1981).

    Google Scholar 

  11. T. R. Stanton and L. A. Ostrovsky, “Observations of highly nonlinear internal solitons over the continental shelf,” Geophys. Res. Lett. 25 (14), 2695–2698 (1998).

    Article  Google Scholar 

  12. K. A. Gorshkov, L. A. Ostrovsky, I. A. Soustova, and V. G. Irisov, “Perturbation theory for kinks and application for multisoliton interactions in hydrodynamics,” Phys. Rev. E 69, 1–10 (2004).

    Article  Google Scholar 

  13. R. Grimshaw, E. Pelinovsky, and T. Talipova, “Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity,” Phys. D (Amsterdam, Neth.) 132, 40–62 (1999).

  14. R. Grimshaw, E. Pelinovsky, and T. Talipova, “Modelling internal solitary waves in the coastal ocean,” Surv. Geophys. 28, 273–287 (2007).

    Article  Google Scholar 

  15. O. Nakoulima, N. Zahybo, E. Pelynovsky, T. Talipova, A. Slunyaev, and A. Kurkin, “Analytical and numerical studies of the variable-coefficient Gardner equation,” Appl. Math. Comput. 152, 449–471 (2004).

    Google Scholar 

  16. O. E. Polukhina and N. M. Samarina, “Cylindrical divergence of solitary internal waves in the context of the generalized Gardner equation,” Izv., Atmos. Ocean. Phys. 43 (6), 755–761 (2007).

    Article  Google Scholar 

  17. C. J. Amick and R. E. L. Turner, “A global theory of internal solitary in two-fluid system,” Trans. Am. Math. Soc. 298, 431–484 (1986).

    Article  Google Scholar 

  18. V. I. Vlasenko, P. Brandt, and A. Rubino, “On the structure of large-amplitude internal solitary waves,” J. Phys. Oceanogr. 30, 2172–2185 (2000).

    Article  Google Scholar 

  19. K. R. Khusnutdinova and X. Zhang, “Long ring waves in stratified fluid over a shear flow,” J. Fluid Mech. 79, 27–44 (2016).

    Google Scholar 

  20. K. R. Khusnutdinova and X. Zhang, “Nonlinear ring waves in two-layer fluid,” Phys. D (Amsterdam, Neth.) 333, 208–220 (2016).

  21. K. A. Gorshkov and I. A. Soustova, “Interaction of solitons as compound structures in the Gardner model,” Radiophys. Quantum Electron. 44 (5), 465–476 (2001).

    Article  Google Scholar 

  22. K. A. Gorshkov, I. A. Soustova, A. V. Ermoshkin, and N. V. Zaitseva, “Evolution of the compound Gardner-equation soliton in the media with variable parameters,” Radiophys. Quantum Electron. 55 (5), 344–355 (2012).

    Article  Google Scholar 

  23. K. A. Gorshkov, I. A. Soustova, A. V. Ermoshkin, and N. V. Zaitseva, “Approximate description of the quasistationary evolution of nearly limiting internal solitary waves using the Gardner equation with variable coefficients,” Fundam. Prikl. Gidrofiz. 6 (3), 54–62 (2013).

    Google Scholar 

  24. K. A. Gorshkov, I. A. Soustova, and A. V. Ermoshkin, “Field structure of a quasisoliton approaching the critical point,” Radiophys. Quantum Electron. 58 (10), 738–744 (2016).

    Article  Google Scholar 

  25. R. Grimshaw, E. Pelinovsky, T. Talipova, and A. Kurkin, “Simulation of the transformation of internal solitary waves on oceanic shelves,” J. Physic. Oceanogr. 34, 2774–2791 (2004).

    Article  Google Scholar 

  26. P. Holloway, E. Pelinovsky, and T. Talipova, “A generalized Korteweg–de Vries model of internal tide transformation in the coastal zone,” J. Geophys. Res. 104 (C8), 18333–18350 (1999).

    Article  Google Scholar 

  27. A. N. Serebryanyi, “Manifestation of soliton properties in internal waves on the shelf,” Izv. Ross. Akad. Nauk: Fiz. Atmos. Okeana 29 (2), 244–252 (1993).

    Google Scholar 

  28. K. D. Sabinin and A. N. Serebryanyi, ““Hot spots” in the field of internal waves in the ocean," Acoust. Phys. 53 (3), 357–380 (2007).

    Article  Google Scholar 

  29. R. Grimshaw, “Initial conditions for the cylindrical Korteweg–de Vries equation,” Stud. Appl. Math. 143 (2), 176–191 (2019).

    Article  Google Scholar 

  30. J. R. Apel, L. A. Ostrovsky, Yu. A. Stepanyants, and J. F. Lynch, “Internal solitons in the ocean and their effect on underwater sound,” J. Acoust. Soc. Am., No. 2, 695–722 (2007).

  31. J. N. Mou, D. M. Farmer, W. D. Smyth, L. Armi, and S. Vagley, “Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf,” J. Phys. Oceanogr. 33, 2093–2112 (2003).

    Article  Google Scholar 

Download references

Funding

This work was carried out within the scope of the state contract for the Institute of Applied Physics, Russian Academy of Sciences, research and development theme no. 0035-2019-0007, and supported in part by the Russian Foundation for Basic Research, project nos. 18-05-00292 and 20-05-00776.

Author information

Authors and Affiliations

  1. Institute of Applied Physics, Russian Academy of Sciences, 603950, Nizhny Novgorod, Russia

    K. A. Gorshkov, L. A. Ostrovsky & I. A. Soustova

Authors
  1. K. A. Gorshkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. A. Ostrovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. A. Soustova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. A. Soustova.

Additional information

Translated by A. Nikolskii

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorshkov, K.A., Ostrovsky, L.A. & Soustova, I.A. Dynamics of Nonstationary Cylindrical Solitary Internal Waves. Izv. Atmos. Ocean. Phys. 57, 170–179 (2021). https://doi.org/10.1134/S0001433821020055

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001433821020055

Keywords:

Search

Navigation