Letters in Mathematical Physics

, Volume 31, Issue 1, pp 1–13 | Cite as

Hamiltonian truncation of the shallow water equation

  • Zhong Ge 
  • Clint Scovel
Article

Abstract

In this Letter, we describe a truncation of the Eulerian description of the shallow water equation of climate modeling to a finite-dimensional Hamiltonian system. The technique is to use an isomorphism from a semidirect product Poisson manifold to a direct product of Poisson manifolds, both of whose components are truncatable to finite-dimensional Poisson manifolds, based on Moser's theorem on volume elements.

Mathematics Subject Classifications (1991)

73D20 35Q35 76B15 65P05 53C80 70H99 

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References

  1. 1.
    Abraham, R. and Marsden, J.,Foundations of Mechanics, 2nd edn, Benjamin/Cummings, Reading, 1978.Google Scholar
  2. 2.
    Benzel, S., Ge, Z., and Scovel, C., Elementary construction of higher order Lie-Poisson integrators,Phys. Lett. A 174, 229–232 (1993).Google Scholar
  3. 3.
    Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions,Comm. Pure Appl. Math. 44, 375–417 (1991).Google Scholar
  4. 4.
    Channell, P. J. and Scovel, J. C., Symplectic integration of Hamiltonian systems,Non-linearity 3, 231–259 (1990).Google Scholar
  5. 5.
    Channell, P. J. and Scovel, J. C., Integrators for Lie-Poisson dynamical systems,Physica D 50, 80–88 (1991).Google Scholar
  6. 6.
    Cheng, S-Y. and Yau, S-T., On the regularity of the Monge-Ampere equation det( 2 u/∂x i ∂x j) =F(x, u),Comm. Pure Appl. Math. 30, 41–68 (1977).Google Scholar
  7. 7.
    Chynoweth, S., The semi-geostrophic equations and the Legendre transform, PhD thesis, University of Reading.Google Scholar
  8. 8.
    Cullen, M. J. P. and Purser, R. J., Properties of the Lagrangian semigeostropic equation,J. Atmos. Sci. 46(17), 2684–2697 (1989).Google Scholar
  9. 9.
    Cullen, M. J. P. and Purser, R. J., An extended Lagrangian theory of semi-geostrophic frontogenesis,J. Atmos. Sci. 41(9), 1477–1497 (1984).Google Scholar
  10. 10.
    Cullen, M. J. P., Norbury, J., and Purser, R. J., Generalized Lagrangian solutions for atmospheric and oceanic flows,SIAM J. Appl. Math. 51(1), 20–31 (1991).Google Scholar
  11. 11.
    Dukowicz, J. K., Cline, M. C., and Addessio, F. L., A general topology Godunov method,J. Comp. Phys. 82(1), 29–63 (1989).Google Scholar
  12. 12.
    Ebin, D. G. and Marsden, J. E., Groups of diffeomorphisms and the motion of an incompressible fluid,Ann. of Math. (2)92, 102–163 (1970); Theory and Lie-Poisson integrators,Phys. Lett. A,133(3), 134–139 (1988).Google Scholar
  13. 13.
    Fairlie, D. B. and Zachos, C. K., Infinite-dimensional algebras, sine brackets, and SU(∞),Phys. Lett. B 224, 101–107 (1989).Google Scholar
  14. 14.
    Feng Kang, Difference scheme for Hamiltonian formalism and symplectic geometry,J. Comput. Math. 4, 27–28 (1986).Google Scholar
  15. 15.
    Feng Kang and Qin Meng-Zhao, The symplectic methods for the computation of Hamiltonian equations, inNumerical Methods for Partial Differential Equations, Lecture Notes in Math. 1297, Springer-Verlag, New York, 1988.Google Scholar
  16. 16.
    Forest, E. and Ruth, R., Fourth-order symplectic integration,Physica D 43, 105–117 (1990).Google Scholar
  17. 17.
    Ge, Z. and Marsden, J., Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,Phys. Lett. A 133, 135–139 (1988).Google Scholar
  18. 18.
    Ge, Z., Equivariant symplectic difference schemes and generating functions,Physica D 49, 376–386 (1991).Google Scholar
  19. 19.
    Guillemin, V. and Sternberg, S.,Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984.Google Scholar
  20. 20.
    Holm, D. D., Kupershmidt, B. A., and Levermore, C. D., Canonical maps between Poisson brackets in eulerian and Lagrangian description of continuum mechanics,Phys. Lett. A 98(8/9), 389–395 (1983).Google Scholar
  21. 21.
    Hoppe, J., Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Reprint of 1982 MIT thesis,Elem. Part. Res. J. (Kyoto)83(3), 145–202 (1989/1990).Google Scholar
  22. 22.
    Krishnaprasad, P. S. and Marsden, J. E., Hamiltonian structures and stability for rigid bodies with flexible attachments,Arch. Rational Mech. Anal. 98, 71–93 (1987).Google Scholar
  23. 23.
    Liao, G., Trong-Whay, P., and Su, J., A numerical grid generator based on Moser's deformation method, preprint (1992).Google Scholar
  24. 24.
    Liao, G. and Su, J., Grid generation via deformation,Appl. Math. Lett. 5(3), 27–29 (1992).Google Scholar
  25. 25.
    Liao, G. and Anderson, D., A new approach to grid generation,Applic. Anal. 44, 285–298 (1992).Google Scholar
  26. 26.
    Marsden, J. and Ratiu, T., Reduction of Poisson manifolds,Lett. Math. Phys. 11, 161–169 (1986).Google Scholar
  27. 27.
    Marsden, J., Ratiu, T., Schmid, R., Spencer, R., and Weinstein, A., Hamiltonian systems with symmetry coadjoint orbits and plasma physics,Proc. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Torino, 7–11 June 1982, Atti della Academia della Scienze di Torino,117, 289–340.Google Scholar
  28. 28.
    Marsden, J. and Ratiu, T.,Mechanics and Symmetry, Springer-Verlag, to appear.Google Scholar
  29. 29.
    Marsden, J., Ratiu, T., and A. Weinstein, Semidirect products and reduction in mechanics,Trans. Am. Math. Soc. 281:1 (1984), 147–176.Google Scholar
  30. 30.
    Montgomery, R., Marsden, J., and T. Ratiu, Gauged Lie-Poisson structures,Cont. Math. AMS 28 (1984), 101–114.Google Scholar
  31. 31.
    Morrison, P. M. and Greene, J. M., Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics,Phys. Rev. Lett. 45, 790–794 (1980).Google Scholar
  32. 32.
    Moser, J., On the volume elements on a manifold,Trans. Amer. Math. Soc. 120, 286–294 (1965).Google Scholar
  33. 33.
    Pedlosky, J.,Geophysical Fluid Dynamics, Springer, New York, 1979.Google Scholar
  34. 34.
    Pogorelov, A. V.,Monge-Ampere Equations of Elliptic Type, Noordhoff, Groningen, 1964.Google Scholar
  35. 35.
    Rouhi, A. and Abarbanel, H. D. I., Symmetric truncations of the shallow water equations, A. Brandt, S. E. Ramberg, and M. F. Shlesinger, (Eds), inNonlinear Dynamics of Ocean Waves, World Scientific, Singapore, 1992.Google Scholar
  36. 36.
    Scovel, C. and Weinstein, A., Finite-dimensional Lie-Poisson approximations to Vlasov-Poisson equations, to appear inComm. Pure Appl. Math. Google Scholar
  37. 37.
    Sanz-Serna, J. M., Runge-Kutta schemes for Hamiltonian systems,BIT 28, 877–883 (1988).Google Scholar
  38. 38.
    Suzuki, M., Fractal decomposition of exponential operators with applications to many-body theories and Monte-Carlo simulations,Phys. Lett. A 146, 319–323 (1990).Google Scholar
  39. 39.
    Yoshida, H., Construction of higher order symplectic integrators,Phys. Lett. A 150, 262–268 (1990).Google Scholar
  40. 40.
    Zeitlin, V., Finite-mode analogues of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure,Physica D 49, 353–362 (1991).Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Zhong Ge 
    • 1
  • Clint Scovel
    • 2
  1. 1.The Fields Institute for Research in Mathematical SciencesOntarioCanada
  2. 2.Los Alamos National LaboratoryComputer Research Group, C-3, MS-B265Los AlamosUSA

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