Duke Mathematical Journal

Semiclassical and spectral analysis of oceanic waves

Christophe Cheverry, Isabelle Gallagher, Thierry Paul, and Laure Saint-Raymond

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In this work we prove that the shallow water flow, subject to strong wind forcing and linearized around an adequate stationary profile, develops for large time closed trajectories due to the propagation of Rossby waves, while Poincaré waves are shown to disperse. The methods used in this paper involve semiclassical analysis and dynamical systems for the study of Rossby waves, while some refined spectral analysis is required for the study of Poincaré waves, due to the large time scale involved which is of diffractive type.

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Duke Math. J., Volume 161, Number 5 (2012), 845-892.

First available in Project Euclid: 27 March 2012

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Zentralblatt MATH identifier

Primary: 35Q86: PDEs in connection with geophysics
Secondary: 76M45: Asymptotic methods, singular perturbations 35S30: Fourier integral operators 81Q20: Semiclassical techniques, including WKB and Maslov methods


Cheverry, Christophe; Gallagher, Isabelle; Paul, Thierry; Saint-Raymond, Laure. Semiclassical and spectral analysis of oceanic waves. Duke Math. J. 161 (2012), no. 5, 845--892. doi:10.1215/00127094-1548407. https://projecteuclid.org/euclid.dmj/1332866805

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  • [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978.
  • [2] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge Univ. Press, Cambridge, 1990.
  • [3] G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn, Boston, 1962.
  • [4] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier, Basics of Mathematical Geophysics, Oxford Univ. Press, New York, 2006.
  • [5] C. Cheverry and T. Paul, On some geometry of propagation in diffractive time scales, Discrete Contin. Dyn. Syst. 32 (2012), 499–538.
  • [6] Y. Colin de Verdière and J. Vey, Le lemme de Morse isochore, Topology 18 (1979), 283–293.
  • [7] M. Dimassi and S. Sjöstrand, Spectral asymptotics in the semi-classical limit; London Math. Soc. Lecture Note Ser. 268, Cambridge Univ. Press, Cambridge, 1999.
  • [8] A. Dutrifoy and A. J. Majda, The dynamics of equatorial long waves: A singular limit with fast variable coefficients, Comm. Math. Sci. 4 (2006), 375–397.
  • [9] A. Dutrifoy and A. J. Majda, Fast wave averaging for the equatorial shallow water equations, Comm. Partial Differential Equations 32 (2007), 1617–1642.
  • [10] A. Dutrifoy, A. J. Majda, and S. Schochet, A simple justification of the singular limit for equatorial shallow-water dynamics, Commun. Pure Appl. Math. 62 (2009), 322–333.
  • [11] I. Gallagher and L. Saint-Raymond, Mathematical study of the betaplane model: Equatorial waves and convergence results, Mém. Soc. Math. Fr. (N.S.) 107 (2006).
  • [12] I. Gallagher and L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, J. Anal. Math. 99 (2006), 1–34.
  • [13] I. Gallagher and L. Saint-Raymond, “On the influence of the Earth’s rotation on geophysical flows” in Handbook of Mathematical Fluid Dynamics, 4. Elsevier, New York, 2007, 201–329.
  • [14] I. Gallagher, T. Paul, and L. Saint-Raymond, “On the propagation of oceanic waves driven by a strong macroscopic flow” in Nonlinear Partial Differential Equations (The Abel Symposium 2010), 7, Springer, Berlin, 231–254, 2012.
  • [15] P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math. 50 (1997), 323–379.
  • [16] A. E. Gill, Atmosphere-Ocean Dynamics, International Geophysics Series 30, 1982.
  • [17] A. E. Gill and M. S. Longuet-Higgins, Resonant interactions between planetary waves, Proc. Roy. Soc. London A 299 (1967), 120–140.
  • [18] H. P. Greenspan, The Theory of Rotating Fluids, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge Univ. Press, Cambridge, 1969.
  • [19] E. Grenier, Oscillatory perturbations of the Navier–Stokes equations, J. Math. Pures Appl. 76 (1997), 477–498.
  • [20] J. Hale and H. Koçak. Dynamics and Bifurcations, Springer, New York, 1991.
  • [21] L. Hörmander, The Analysis of Linear Operators, Vol. III, Springer, New York, 1985.
  • [22] J. H. Hubbard and B. H. West, Differential Equations: A Dynamical System Approach, II, Texts Appl. Math., Springer, New York, 1995.
  • [23] J.-L. Joly, G. Métivier, and J. Rauch, Coherent nonlinear waves and the Wiener algebra, Ann. Inst. Fourier (Grenoble) 44 (1994), 167–196.
  • [24] A. Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lecture Notes in Math. 9, Amer. Math. Soc, Providence, 2003.
  • [25] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Springer, New York, 2002.
  • [26] J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979.
  • [27] J. Pedlosky, Ocean Circulation Theory, Springer, New York, 1996.
  • [28] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations 114 (1994), 476–512.
  • [29] E. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton, 1993.
  • [30] S. Vũ Ngoc, Systèmes intégrables semi-classiques: du local au global, Panor. Synthèses 22, Soc. Math. France, Montrouge, 2006.