MOTION OF FLUID PARTICLES IN THE FIELD OF A NONLINEAR PERIODIC SURFACEWAVE IN A FLUID BENEATH AN ICE COVER

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

A finite-depth fluid layer described by the Euler equations is considered. The ice cover is simulated by a geometrically nonlinear elastic Kirchhoff–Love plate. The trajectories of fluid particles under the ice cover are in the field of nonlinear surface periodic traveling waves of small, but finite amplitude. A solution describing such surface waves is allowed by the equations of the model. Periodic waves are described by Jacobi elliptic functions. The analysis uses explicit asymptotic expressions for solutions describing wave structures at the water–ice interface, such as a periodic wave against a zero-displacement surface, as well as asymptotic solutions for the velocity field in a fluid column generated by these waves.

About the authors

A. T. Il’ichev

Steklov Mathematical Institute, Russian Academy of Sciences; Bauman Moscow State Technical University

Email: ilichev@mi-ras.ru
Moscow, Russia; Moscow, Russia

A. S. Savin

Bauman Moscow State Technical University; Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences

Moscow, Russia; Moscow, Russia

References

  1. Forbes L.K. Surface waves of large amplitude beneath an elastic sheet. High order series solution // J. Fluid Mech. 1986. V. 169. P. 409–428.
  2. Forbes L.K. Surface waves of large amplitude beneath an elastic sheet. Galerkin solutions // J. Fluid Mech. 1988. V. 188. P. 491–508.
  3. Kirchgassner K. Wave solutions of reversible systems and applications // J. Diff. Eqns. 1982. V. 45. P. 113–127.
  4. Mielke A. Reduction of quasilinear elliptic equations in cylindrical domains with applications // Math. Meth. Appl. Sci. 1988. V. 10. P. 501–566.
  5. Iooss G., Kirchgassner K. Water waves for small surface tension:an approach via normal form // Proc. Roy. Soc. Edinburgh. 1992. V. 122A. P. 287–299.
  6. Iooss G., Adelmeyer M. Topics in bifurcation theory and applications. 2nd ed. Singapore: World Sci., 1998.
  7. Ильичев А.Т. Уединенные волны в средах с дисперсией и диссипацией (обзор) // Изв. РАН,МЖГ. 2000.№2. С. 3–27.
  8. Parau E., Dias F. Nonlinear effects in the response of a floating ice plate to a moving. load // J. Fluid. Mech. 2001. V. 37. P. 325–336.
  9. Plotnikov P.I., Toland J.F. Modelling nonlinear hydroelastic waves // Phil. Trans. R. Soc. A. 2011. V. 369. P. 2942–2956.
  10. Ильичев А.Т., Савин А.С., Шашков А.Ю. Траектории частиц жидкости под ледяным покровом в поле уединенной изгибно-гравитационной волны // Изв. вузов. Радиофизика. 2023. Т. 66.№10. С. 848–861.
  11. Филлипс О.М. Динамика верхнего слоя океана. Л.: Гидрометеоиздат, 1980. 320 с.
  12. Монин А.С. Теоретические основы геофизической гидродинамики. Л.: Гидрометеоиздат, 1988. 425 с.
  13. Stokes G. On the theory of oscillatory waves // Trans. Camb. Phil. Soc. 1947. V. 8. P. 314–326.
  14. Очиров А.А. Исследование закономерностей формирования массопереноса, инициируемого волновыми движениями жидкости: диссерт. на соискание степени кандидата физ.-мат. наук. Ярославль, 2020. 142 с.
  15. Ильичев А.Т, Савин А.С., Шашков А.Ю. Траектории жидких частиц в поле темного солитона в жидкости под ледяным покровом // Изв. РАН. МЖГ. 2023.№6. С. 110—120.
  16. Ильичев А.Т., Савин А.С., Шашков А.Ю. Движение частиц в поле нелинейных волновых пакетов в слое жидкости под ледяным покровом // Теор. и матем. физ. 2024. Т. 218. С. 586–600.
  17. Il’ichev A.T, Savin A.S., Shashkov A.Yu. Motion of liquid particles in the field of 1:1 resonanse nonlinear wave structures in a fluid beneath an ice cover // Int. J. Non-Linear Mech. 2024. V. 160. P. 104665.
  18. Muller A., Ettema R. Dynamic response of an icebreaker hull to ice breaking // In Proc. IAHR Ice Symp., Hamburg. 1984. II. P. 287–296.
  19. Ильичев А.Т. Уединенные волны в моделях гидромеханики. М.: Физматлит, 2003.
  20. Ильичев А.Т. Солитоноподобные структуры на поверхности раздела вода–лед // Успехи матем. наук. 2015. Т. 70. С. 85–138.
  21. Haragus M., Iooss G. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. Berlin, Heidelberg: Springer, 2012. 329 p.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2025 Russian Academy of Sciences