Advertisement

The Role of the Earth’s Rotation: Oscillations in Semi-bounded and Bounded Basins of Constant Depth

  • Kolumban HutterEmail author
  • Yongqi Wang
  • Irina P. Chubarenko
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM, volume 2)

Abstract

In Chap. 7 of Volume I, the propagation of surface waves in a layer of a homogeneous fluid referred to an inertial frame was studied. It was shown that superposing the fields of two waves, with the same frequency propagating in opposite directions with the same amplitude can be combined to a standing wave. These standing waves appear as localized oscillations between fixed nodal lines of which the distance defines the semi-wave length with wave humps and wave troughs arising inbetween. Under frictionless conditions imaginary walls can be placed at any position parallel to the wave direction to confine a channel without physically violating any boundary conditions. Similarly, the locations of the nodal lines across the channel turned out to be the positions of standing waves where the longitudinal velocity component vanishes for all time so that vertical walls can equally be inserted at these positions without disturbing the solution. This then formally yields the surface wave solution for the unidirectional motion in a basin of rectangular form and constant depth, see Figs. 7.9 and 7.12 in Chap. 7 of Volume I. These standing wave solutions were subsequently generalized to two-dimensional oscillations in rectangular cells of constant depth in which non-vanishing horizontal velocity components are allowed within the cell that only persistently vanish at the four side walls, thus forming oscillations of true cellular structure (see Figs. 7.14 and 7.15 in Chap. 7 in Volume I). How does the structure of these waves change when the fluid is rotating?

Keywords

Wave Solution Gravity Wave Constant Depth Baroclinic Mode Homogeneous Fluid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bjerknes, V., Bjerknes, J., Solberg, H. and Bergeron, T.T.: Physikalische Hydrodynamik, Springer, Berlin, etc. (1933)Google Scholar
  2. 2.
    Brown, P.J.: Kelvin wave reflection in a semi-infinite canal. J. Mar. Res., 31, 1–10 (1973)Google Scholar
  3. 3.
    Bryan, G.: The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. Royal Soc. Lond., 180, 187–219 (1889)CrossRefGoogle Scholar
  4. 4.
    Cartwright, D.: Tides, a Scientific History. Cambridge University Press, Cambridge (1999)Google Scholar
  5. 5.
    Chapman, D.: On the failure of Laplace’s tidal equations to model sub-inertial motions at a discontinuity in depth. Dyn. Atmos. Oceans, 7, 1–16 (1982)CrossRefGoogle Scholar
  6. 6.
    Chapman, D. and Hendershott, M.: Shelf wave dispersion in a geophysical ocean. Dyn Atmos. Oceans, 7, 17–31 (1982)CrossRefGoogle Scholar
  7. 7.
    Corkan, R.H. and Doodson, A.T.: Free oscillations in a rotating square sea. Proc. Roy. Soc. London A, 21, 147 (1952)CrossRefGoogle Scholar
  8. 8.
    Defant, A.: Gezeitenprobleme des Meeres in Landnähe. (published in “Probleme der kosmischen Physik”, Vol. 6) Hamburg, Henri Grand, p 80 (1925)Google Scholar
  9. 9.
    Defant, F.: Theorie der Seiches des Michigansees und ihre Abwandlung durch Wirkung der Corioliskraft. Arch. Met. Geophys. Biokl., A 6, 218–241 (1953)Google Scholar
  10. 10.
    Goldsbrough, G.R.: The tidal oscillations in rectangular basins. Proc. Roy. Soc. London A, 132, 689 (1931)CrossRefGoogle Scholar
  11. 11.
    Jeffreys, H.: The free oscillations of water in an elliptical lake. Proc. Lond. Math. Soc., 23, 455–476 (1925)CrossRefGoogle Scholar
  12. 12.
    Kelvin, L. (William Thomson): On the gravitational oscillations of rotating water. Proc. Roy. Soc. Edinburgh, 10, 92–100, (1979); reprinted Phil. Mag., 10, 109–116 (1880)Google Scholar
  13. 13.
    Kelvin, L.: Vibrations of a columnar vortex. Philos. Mag., 10, 155–168 (1880)CrossRefGoogle Scholar
  14. 14.
    Krauss, W.: Interne Wellen. Gebrüder Bornträger, Berlin, Nikolassee (1966)Google Scholar
  15. 15.
    Krauss, W.: Methods and Results of Theoretical Oceanography I, Dynamics of the homogeneous and the quasi-homogeneous Ocean. Gebr. Bornträger (1973)Google Scholar
  16. 16.
    Lamb, H.: Hydrodynamics. Cambridge University Press (1924)Google Scholar
  17. 17.
    Lamb, H.: Hydrodynamics. 6th edition, Cambridge University Press (1932)Google Scholar
  18. 18.
    LeBlond, P.H. and Mysak, L. A.: Waves in the Ocean Elsevier Sci. Publ, Amsterdam 602 p.(1978)Google Scholar
  19. 19.
    Lifschitz, A. and Fabijonas, B.: A new class of instabilities of rotating fluids. Phys. Fluids, 8, 2239–2241 (1996)CrossRefGoogle Scholar
  20. 20.
    Lighthill, S.M.J.: Dynamics of rotating fluids: a survey. J. Fluid Mech., 26, 411–431 (1966)CrossRefGoogle Scholar
  21. 21.
    Maas, L.R.M.: On the amphidromic structure of inertial waves in a rectangular parallelepiped. Fluid Dynamics Research, 33, 373–401 (2003)CrossRefGoogle Scholar
  22. 22.
    Mortimer, C. H.: Frontiers in physical limnology with particular reference to long waves in rotating basins. Proc. 5 th Conf. Great Lakes Res., Great Lakes Res. Div. Univ. Michigan, 9, 9–42 (1963)Google Scholar
  23. 23.
    Mortimer, C. H.: Large scale oscillatory motions and seasonal temperature changes in Lake Michigan and Lake Ontario Special Report No 12, Part I, 11 p.: Text, Part II, 106 p.: Illustrations, Center for Great Lakes Studies, The University of Wisconsin-Milwaukee (1971)Google Scholar
  24. 24.
    Mortimer, C. H.: Lake Hydrodynamics Mitt. Int. Ver. Theor. Angew. Limnol., 20, 124–197 (1974)Google Scholar
  25. 25.
    Mortimer, C. H.: Substantive corrections to SIL Communications (IVL Mitteilungen) Nrs 6 and 20. Mitt. Int. Ver. Theor. Angew. Limnol., 19, 60–72 (1975)Google Scholar
  26. 26.
    Mortimer, C. H.: Internal waves observed in Lake Ontario during the International Field Year for the Great Lakes (IFYGL), 1977: descriptive theory and preliminary interpretations of near-inertial oscillations in terms of linear channel models. Special Report no 32, Center for Great Lakes Studies, The University of Wisconsin-Milwaukee, 122 p. (1977)Google Scholar
  27. 27.
    Mortimer, C. H.: Internal motion and related internal waves in Lake Michigan and Lake Ontario as responses to impulsive wind stresses. Special Report no 37, Center for Great Lakes Studies, The University of Wisconsin-Milwaukee, 192 p. (1980)Google Scholar
  28. 28.
    Platzman, G. and Rao, D.B.: The free oscillations of Lake Erie. In: Studies on Oceanography (Hidaka Volume) (ed. K. Yoshida), 359–382, University of Washington Press (1964)Google Scholar
  29. 29.
    Platzman, G. : Ocean Tides and Related Waves Amer. Math. Soc. Lectures in Applied Mathematics, 14, 239–291 (1971)Google Scholar
  30. 30.
    Poincaré, H.: Sur l’équilibre d’une masse de fluide animee d’un movement de rotation. Acta Mathematica, VII, 259–380 (1885)Google Scholar
  31. 31.
    Poincaré, H.: Sur la precession des corps deformables. Bull. Astronom., 27, 321 (1910)Google Scholar
  32. 32.
    Prandle, D.: The vertical structure of tidal currents. Geophys. Astrophys. Fluid Dyn., 22, 29–49 (1982)CrossRefGoogle Scholar
  33. 33.
    Proudman, J.: On the dynamical theory of tides. Part (ii): flat seas. Proc. London Math. Soc., 2nd Series, 18, 21–50 (1916)Google Scholar
  34. 34.
    Proudman, J.: Note on the free tidal oscillations of a sea with slow rotation. Proc. Lond. Math. Soc., (2nd series), 35, 75 (1933)Google Scholar
  35. 35.
    Raggio, G. and Hutter, K.: An extended channel model for the prediction of motion in elongated homogeneous lakes. Part 1. Theoretical introduction. J. Fluid Mech., 121, 231-255 (1982)Google Scholar
  36. 36.
    Raggio, G. and Hutter, K.: An extended channel model for the prediction of motion in elongated homogeneous lakes. Part 2. First order model applied to ideal geometry: rectangular basins with flat bottom. J. Fluid Mech., 121, 257-281 (1982)Google Scholar
  37. 37.
    Raggio, G. and Hutter, K.: An extended channel model for the prediction of motion in elongated homogeneous lakes. Part 3. Free oscillations in natural basins J. Fluid Mech., 121, 283-299 (1982)Google Scholar
  38. 38.
    Hutter, K. and Raggio, G.: A Chrystal-model describing gravitational barotropic motion in elongated lakes Arch. Met. Geophys. Biokl., Ser A, 31, 361-378 (1982)Google Scholar
  39. 39.
    Rao, D.B.: Free gravitational oscillations in rotating rectangular basins. Ph.D Thesis, Department of Geophysical Sciences, The University of Chicago (1965)Google Scholar
  40. 40.
    Rao, D.B.: Free gravitational oscillations in rotating rectangular basins. J. Fluid Mech., 25, 523–555 (1966)CrossRefGoogle Scholar
  41. 41.
    Rayleigh, L.: On the vibrations of a rectangular sheet of rotating liquid. Phil. Mag., 5, 297 (1903)CrossRefGoogle Scholar
  42. 42.
    Rayleigh, L.: Notes concerning tidal oscillations upon a rotating globe. Proc. Roy. Soc. A, 82, 448 (1909)CrossRefGoogle Scholar
  43. 43.
    Rieutord, M., Georgeot, B. and Valdettaro, L.: Wave attractors in rotating fluids: a paradigm for ill-posed Cauchy problems. Phys. Rev. Lett., 85, 4277–4280, (2000)CrossRefGoogle Scholar
  44. 44.
    Solberg, H.: Über die freien Schwingungen einer homogenen Flüssigkeitsschicht auf der rotierenden Erde. I. Astophys. Norv., 1, 237–340 (1936)Google Scholar
  45. 45.
    Taylor, G.I.: Tidal oscillations in gulfs and basins. Proc. London Math. Soc., Series 2, XX, 148–181 (1920)Google Scholar
  46. 46.
    Van Danzig, D. and Lauwerier, H.A.: The North Sea Problem. IV. Free oscillations of a rotating rectangular sea. Proc. K. ned. Akad. Wet. (Series A), 63, 339 (1960)Google Scholar
  47. 47.
    Whewell: Essay towards a first approximation to a map of co-tidal lines. Phil. Trans. Royal Soc. London, 123, 147–236 (1833)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kolumban Hutter
    • 1
    Email author
  • Yongqi Wang
    • 2
  • Irina P. Chubarenko
    • 3
  1. 1.c/o Versuchsanstalt für Wasserbau Hydrologie und Glaziologie ETH-ZentrumETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringDarmstadt University of TechnologyDarmstadtGermany
  3. 3.P.P. Shirshov Institute of OceanologyRussian Academy of SciencesKaliningradRussia

Personalised recommendations