Success of Arnol’d’s Method in a Hierarchy of Ocean Models

Conference paper
Part of the Springer Series in Nonlinear Dynamics book series (SSNONLINEAR)


The method of Arnol’d[l, 2] is used to derive stability conditions of Hamiltonian systems with singular Poisson brackets; notably, those of fluid mechanics:[3, 4, 5, 6, 7] Let the field(s) φ(x, t) represent the state variables, and the Hamiltonian H [φ] and Poisson bracket {. , .} be such that ∂t φ = {φ, H}. The bracket must have the usual properties;[5] in addition, it may be singular, i.e. there may exist non-trivial functional(s) of state C[φ]-called Casimir(s)- such that {C, F} a ≡ 0, for any admissible[8] functional of state F[φ].


Vortex Manifold Stratification Vorticity Weinstein 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References & Footnotes

  1. [1]
    Arnold, V.I., 1965. Condition for nonlinear stationary plane curvilinear flows of an ideal fluid, Dokl. Akad. Nauk. USSR, 162: 975–978; (English transl: Soviet Math., 6: 773–777, 1965).Google Scholar
  2. [2]
    Arnold, V.I., 1966. On an apriori estimate in the theory of hydrodynamical stability, Izv. Vyssh. Uchebn. Zaved. Matematika, 54: 3–5; (English transl. Amer. Math. Soc. Transi., Series 2 79: 267–269, 1969).Google Scholar
  3. [3]
    Holm, D.D., J.E. Marsden, T. Ratiu, and A. Weinsten, 1985. Nonlinear stability of fluid and plasma equilibria, Phys. Reports, 123: 1–116.ADSMATHCrossRefGoogle Scholar
  4. [4]
    Abarbanel, H.D.I., D.D. Holm, J.E. Marsden and T. Ratiu, 1986. Nonlinear stability of stratified fluid equilibria, Phil. Trans. Roy. Soc. London, A 318: 349–409.MathSciNetADSGoogle Scholar
  5. [5]
    McIntyre, M.E., and T.G. Shepherd, 1987. An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol’d’s stability theorems, J. Fluid Mech., 181: 527–565.MathSciNetADSMATHCrossRefGoogle Scholar
  6. [6]
    Shepherd, T.G., 1990. Symmetries, conservation laws, and hamiltonian structure in geophysical fluid dynamics, Adv. Geophys., 32: 287–338.ADSCrossRefGoogle Scholar
  7. [7]
    Shepherd, T.G., 1990. Arnol’d stability applied to fluid flow: successes and failures, in Nonlinear phenomena in atmospheric and oceanic sciences, G.F. Carnevale & R.T. Pierrehumbert, eds.Google Scholar
  8. [9]
    Shepherd, T.G., 1988. Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows, J. Fluid Mech., 196: 291–322.MathSciNetADSMATHCrossRefGoogle Scholar
  9. [10]
    Shepherd, T.G., 1988. On the nonlinear saturation of baroclinic instability. Part I: The two layer model, J. Atmos. Sci., 45: 2014–2025.MathSciNetADSCrossRefGoogle Scholar
  10. [11]
    Shepherd, T.G., 1989. Nonlinear saturation of baroclinic instability. Part II: continuously stratified fluid, J. Atmos. Sci., 46: 888–907.MathSciNetADSCrossRefGoogle Scholar
  11. [12]
    Andrews, D.G., 1984. On the existence of nonzonal flows satisfying sufficient conditions for stability, Geophys. Astrophys. Fluid Dyn., 28: 243–256.ADSCrossRefGoogle Scholar
  12. [13]
    Carnevale, G.F., and T.G. Shepherd, 1990. On the interpretation of Andrews’ theorem, Geophys. Astrophys. Fluid Dyn., 51: 1–17.MathSciNetADSCrossRefGoogle Scholar
  13. [14]
    Ripa, P., 1991. A tale of three theorems, Rev. Méx. Fis., submitted.Google Scholar
  14. [15]
    Ripa, P., 1991. General stability conditions for a multi-layer model, J. Fluid Mech., 222: 119–137.MathSciNetADSMATHCrossRefGoogle Scholar
  15. [16]
    Ripa, P., 1991. Wave energy-momentum and pseudo energy-momentum conservation for the layered quasi-geostrophic instability problem, J. Fluid Mech., in press.Google Scholar
  16. [17]
    Blumen, W., 1968. On the Stability of Quasi-Geostrophic Flow, J. Atmos. Sci., 25: 929–931.ADSCrossRefGoogle Scholar
  17. [18]
    Salusti, E., and F. Zirilli, 1986. A Large class of Non-zonal oceanic flows satisfying the Arnold-Blumen sufficient condition for Stability, Geophys. Astrophys. Fluid Dyn., 35: 157–171.ADSMATHCrossRefGoogle Scholar
  18. [19]
    Swaters, G.E., 1986. A nonlinear stability theorem for baroclinic quasigeostrophic flow, Phys. Fluids, 29: 5–6.MathSciNetADSMATHCrossRefGoogle Scholar
  19. [20]
    Blumen, W., 1973. Stability of a two-layer fluid model to nongeostrophic disturbances, Tellus, 25: 12–19.ADSCrossRefGoogle Scholar
  20. [21]
    Holm, D.D., J.E. Marsden, T. Ratiu, and A. Weinstein, 1983. Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics, Phys. Lett., 98 A: 15–21.MathSciNetADSGoogle Scholar
  21. [22]
    Ripa, P., 1987. On the stability of elliptical vortex solution of the shallow-water equations, J. Fluid. Mech., 183: 343–363.ADSMATHCrossRefGoogle Scholar
  22. [23]
    Ripa, P., 1990. Positive, negative and zero wave energy and the flow stability problem, in the Eulerian and LagrangianEulerian descriptions, Pure Appl. Geophys., 133: 713–732.ADSCrossRefGoogle Scholar
  23. [24]
    Miles, J.W., 1961. On the stability of heterogeneous shear flows, J. Fluid Mech., 10: 496.MathSciNetADSMATHCrossRefGoogle Scholar
  24. [25]
    Howard, L.N., 1961. Note on paper of John W. Miles, J. Fluid Mech., 10: 509.MathSciNetADSMATHCrossRefGoogle Scholar
  25. [26]
    Benzi, R., S. Pierini, A. Vulpiani, and E. Salusti, 1982. On nonlinear hydrodynamic stability of planetary vortices, Geophys. Astrophys. Fluid Dyn., 20: 293–306.ADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • P. Ripa
    • 1
  1. 1.C.I.C.E.S.E.EnsenadaMéxico

Personalised recommendations