Nonlinear effects at propagation of long surface waves in the channels with a variable cross-section

A.Yu. Bazykina, S.F. Dotsenko✉

Marine Hydrophysical Institute, Russian Academy of Sciences, Sevastopol, Russian Federation

✉ e-mail: sf_dotsenko@mail.ru

Abstract

Analysis of the propagation of waves in the channels and straits of different geometry seems an important problem, since the change of the channel width, its depth and cross-sectional shape in general significantly affects the amplification and weakening of the spatial structure of the wave field. Propagation and deformation of single surface waves in the channels with a variable cross-section are analyzed within the framework of shallow-water equations. It is shown that nonlinearity is manifested in growth with time of the wave front slope steepness and its subsequent breaking. Height and length of a propagating wave are weakly influenced by nonlinearity. The distance traversed by a single wave up to its breaking decreases with growth of the wave height and diminution of its length. The wave amplitude characteristics are estimated depending on the channel depth and width. They are accurately described by the known Green's law. The propagation of long waves in channels with different cross-sectional shape at the same maximum depth and width had been numerically analyzed. The strongest nonlinearity was detected in the channel with a triangular cross section, as in this case the cross-sectional area was the least with the same depth and width. The channel shape has no significant effect on the amplitude and wave characteristics.

Keywords

waves in fluid, waves of finite amplitude, wave propagation in channels, channel mathematical model, numerical solutions, manifestation of nonlinear effects

For citation

Bazykina, A.Yu. and Dotsenko, S.F., 2015. Nonlinear effects at propagation of long surface waves in the channels with a variable cross-section. Physical Oceanography, (4), pp. 3-12. doi:10.22449/1573-160X-2015-4-3-12

DOI

10.22449/1573-160X-2015-4-3-12

References

  1. Voltsinger, N.E., Pyaskovsky, R.V., 1968, “Osnovnye okeanologicheskie zadachi teorii melkoj vody [Basic oceanographic problems in the theory of shallow water]”, Leningrad, Gidrometeoizdat, 300 p. (in Russian).
  2. Cherkesov, L.V., 1976, “Gidrodinamika poverkhnostnykh i vnutrennikh voln [Hydrodynamics of surface and internal waves]”, Kiev, Naukova Dumka, 264 p. (in Russian).
  3. Pelinovsky, E.N., 1982, “Nelineynaya dinamika voln tsunami [Nonlinear dynamics of tsunami waves]”, Gorky, Institut Prikladnoy Fiziki AN SSSR, 226 p. (in Russian).
  4. Marchuk, A.G., Chubarov, L.B. & Shokin, Yu.I., 1983, “Chislennoe modelirovanie voln tsunami [Numerical modeling of tsunami waves]”, Novosibirsk, Nauka, 175 p. (in Russian).
  5. Selezov, I.T., Sidorchuk, V.N. & Yakovlev, V.V., 1983, “Transformatsiya voln v pribrezhnoy zone shelfa [Transformation of waves in the coastal zone of a shelf]”, Kiev, Naukova Dumka, 207 p. (in Russian).
  6. Kononkova, G.E., Pokazeev, K.V., 1985, “Dinamika morskikh voln [Sea wave dynamics]”, Moscow, Moscow State University Publishers, 298 p. (in Russian).
  7. Voltsinger, N.E., Klevanny, K.A. & Pelinovsky, E.N., 1989, “Dlinnovolnovaya dinamika pribregnoj zony [Long-wave dynamics of the coastal zone]”, Leningrad, Gidrometeoizdat, 271 p. (in Russian).
  8. Shokin, Yu.I., Chubarov, L.B. & Marchuk, A.G. [еt al.], 1989, “Vychislitelnyj experiment v probleme tsunami [Numerical experiment in tsunami problem]”, Novosibirsk, Nauka, 168 p. (in Russian).
  9. Pelinovsky, E.N., 1996, “Gidrodinamika voln tsunami [Hydrodynamics of tsunami waves]”, N. Novgorod, Institut Prikladnoy Fiziki RAN Press, 276 p. (in Russian).
  10. Kurkin, A.A., 2005, “Nelineinaya i nestatsionarnaya dinamika dlinnykh voln v pribrezhnoy zone [Nonlinear and nonstationary dynamics of long waves in the coastal zone]”, N. Novgorod, Nizhegor. Cos. Tech. Univ., 330 p. (in Russian).
  11. Levin, B.V., Nosov, M.A., 2005, “Fizika tsunami [Physics of Tsunamis]”, Moscow, Yanus-K, 360 p. (in Russian).
  12. Didenkulova, I.I., Pelinovsky, D.E. & Tyugin, D.Y. [еt al.], 2012, “Begushchie dlinnye volny v vodnyh pryamougolnyh kanalah peremennogo secheniya [The running long waves in the water rectangular channels of variable section]”, Vestnik MGRU, no. 5, p. 89 – 93 (in Russian).
  13. Dotsenko, S.F., Rakova, I.N., 2012, “Rasprostranenie dlinnykh poverkhnostnykh voln v kanalakh peremennogo poperechnogo secheniya [Propagation of long surface waves in the channels of variable cross-section]”, Morskoy gidrofizicheswkiy zhournal, no. 2, p. 3 – 17 (in Russian).
  14. Bazykina, A.Yu., Dotsenko, S.F., 2015, “Primenenie kanalovoy modeli dlya opisaniya rasprostraneniya odinochnykh voln tipa tsunami v kanale peremennogo poperechnogo secheniya [Application of a channel model for describing propagation of tsunami-like single waves in the variable cross-section channel]”, Morskoy gidrofizicheswkiy zhournal, no.1, p. 29 – 41 (in Russian).
  15. Pelinovsky, E.N., Troshina, E.N., 1993, “Rasprostranenie dlinnykh voln v prolivakh [Propagation of long waves in straits]”, Morskoy gidrofizicheswkiy zhournal, no.1, p. 47 – 52 (in Russian).
  16. Petrukhin, N.S., Pelinovsky, E.N., 2011, “Modelirovanie vodnogo potoka v ramkakh odnomernykh uravneniy melkoy vody [Water flow modeling in the framework of 1d shallow-water equations]”, Trudy NGTU, no. 4 (91), p. 60 – 69 (in Russian).
  17. Didenkulova I.I., Pelinovsky E.N., 2011, “Runup of tsunami waves in U-shaped bays”, Pure Appl. Geophys., no.168, p. 1239 – 1249.
  18. Didenkulov, O.I., Didenkulova, I.I., Pelinovsky, E.N., 2014, “Parametrizatsiya kharakteristik nakata odinochnykh voln v bukhte parabolicheskogo secheniya [Parameterization characteristics of solitary waves in the bay of parabolic cross-section]”, Trudy NGTU, no. 3 (105), p. 30 – 36.
  19. Knysh, V.V., 1971, “O nelineinykh effektakh pasprostraneniya dlinnykh voln tipa tsunami [On nonlinear effects of propagation of long waves of tsunami type]”, Morskie gidrofizicheskie issledovaniya, no. 6 (56), p. 15 – 25 (in Russian).
  20. Lamb, G., 1947, “Gidrodinamika [Hydrodynamics]”, Moscow, Leningrad, Gostekhizdat, 928 p. (in Russian).
  21. Friedrichs, C.T., Aubrey, D.G., 1994, “Tidal propagation in strongly convergent channels”, J Geoph. Res., vol. 99, no. C2, p. 3321 – 3336.

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