Gravity Waves on the Surface of the Sphere

  • R. de la Llave
  • P. Panayotaros
Conference paper

Summary

We propose a Hamiltonian model for gravity waves on the surface of a fluid layer surrounding a gravitating sphere. The general equations of motion are nonlocal and can be used as a starting point for simpler models, which can be derived systematically by expanding the Hamiltonian in dimensionless parameters. In this paper, we focus on the small wave amplitude regime. The first-order nonlinear terms can be eliminated by a formal canonical transformation. Similarly, many of the second order terms can be eliminated. The resulting model has the feature that it leaves invariant several finite-dimensional subspaces on which the motion is integrable.

Keywords

Vortex Manifold Vorticity Azimuth Alan 

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References

  1. [B]
    T.B. Benjamin, P.J. Olver: Hamiltonian structure, symmetries and conservation laws for water waves, J. Fluid Mech., 83, 137–185 (1982).ADSCrossRefMathSciNetGoogle Scholar
  2. [BL]
    L. C. Biedenharn, J. D. Louck: Angular momentum in quantum physics—theory and applications, Addison-Wesley, Reading, Mass. (1981).Google Scholar
  3. [C]
    J. R. Cary: Lie transform perturbation theory for Hamiltonian systems, Phys. Reports, 79, No. 2, 129 – 159 (1981).ADSCrossRefMathSciNetGoogle Scholar
  4. [CM]
    R. R. Coifman, Y. Meyer: Non-linear harmonic analysis and analytic dependence, AMS Proc. Symp. Pure Math., 43, 71 – 78 (1985).MathSciNetGoogle Scholar
  5. [CG]
    W. Craig, M. D. Groves: Hamiltonian long-wave scaling limits of the water wave problem, preprint (1992).Google Scholar
  6. [CS]
    W. Craig, C. Sulem: Numerical simulation of gravity waves, J. Comp. Phys., 108, 73 – 83 (1993).ADSMATHCrossRefMathSciNetGoogle Scholar
  7. [DF]
    A. J. Dragt, J. M. Finn: Lie series and invariant functions for analytic symplectic maps, J. Math. Phys., 17, 2215 – 2227 (1976).ADSMATHCrossRefMathSciNetGoogle Scholar
  8. [D]
    B. A. Dubrovin: Geometry of Hamiltonian evolutionary systems, Bibliopolis, Napoli (1991).MATHGoogle Scholar
  9. [FS]
    Z. C. Feng, P. R. Sethna: Symmetry breaking and bifurcations in resonant surface waves, J. Fluid Mech., 199, 495–518 (1989).ADSCrossRefMathSciNetGoogle Scholar
  10. [GAS]
    M. Glozman, Y. Agnon, M. Stiassnie: High order formulation of the water wave problem, Physica D, 66, 347 – 367 (1993).ADSMATHCrossRefMathSciNetGoogle Scholar
  11. [LL]
    L. D. Landau, A. Lifschitz: Fluid Mechanics, Pergamon, Oxford (1959).Google Scholar
  12. [L]
    H. Lamb: Hydrodynamics, Dover, New York (1932).MATHGoogle Scholar
  13. [LMMR]
    D. Lewis, J. Marsden, R. Montgomery, T. Ratiu: The Hamiltonian structure for dynamic free boundary problems, Physica D, 18, 391 – 404 (1986).ADSMATHCrossRefMathSciNetGoogle Scholar
  14. [M]
    J. W. Miles: On Hamilton’s principle for surface waves, J. Fluid Mech., 83, 153 – 158 (1977).ADSMATHCrossRefMathSciNetGoogle Scholar
  15. [MO]
    J. Moser: Periodic orbits near an equilibrium and a theorem of Alan Weinstein, Commun. Pure Appl. Math., 29, 727 – 747 (1976).ADSMATHCrossRefGoogle Scholar
  16. [MG]
    P. M. Morrison, J. M. Greene: Non-canonical Hamiltonian density formulation of hydrodynamics and ideal magneto-hydrodynamics, Phys. Rev Let., 45, 790 – 794 (1980).ADSCrossRefMathSciNetGoogle Scholar
  17. [MW]
    J. Marsden, A. Weinstein: Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica D, 7, 305 – 323 (1983).ADSCrossRefMathSciNetGoogle Scholar
  18. [McR]
    T. M. MacRoberts: Spherical harmonics, Pergamon Press, London, (1967).Google Scholar
  19. [P]
    G. W. Platzman: Ocean tides and related waves, in Mathematical Problems in the Geophysical Sciences I, ed. W. H. Reid, 239–291 (1970).Google Scholar
  20. [We]
    A. Weinstein: Normal modes for non-linear Hamiltonian systems, Inv. Math., 20, 47 – 57 (1973).ADSMATHCrossRefGoogle Scholar
  21. [wi]
    G. B. Whitham: Linear and Nonlinear Waves, Wiley Interscience, New York (1974).MATHGoogle Scholar
  22. [Z]
    V. E. Zakharov: Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Tech. Phys., 2, 190 – 194 (1968).ADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • R. de la Llave
    • 1
  • P. Panayotaros
    • 2
  1. 1.Department of MathematicsThe University of Texas at AustinAustinUSA
  2. 2.Department of PhysicsThe University of Texas at AustinAustinUSA

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