Asymptotics of a slow manifold

SIAM Journal on Applied Dynamical Systems

Volumn 7, Issue 4, 2008, Pages 1163-1190

  Vanneste, J ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Atmospheric waves; Exponential asymptotics; Slow manifold

Indexed keywords

MOTION ESTIMATION;

ASYMPTOTIC PROCEDURES; ASYMPTOTICS; ATMOSPHERIC WAVES; BOREL SUMMATIONS; EXPONENTIAL ASYMPTOTICS; FAST OSCILLATIONS; PIECE WISES; SERIES EXPANSIONS; SLOW MANIFOLD; SLOW MANIFOLDS; SWITCH ON; TIME SCALES;

ASYMPTOTIC ANALYSIS;

EID: 56449106366     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/070710081     Document Type: Article

References (35)

1. M. ADRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions, Dover, New York, 1965.
2. W. BALSER, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Universitext, Springer-Verlag, New York, 2000.
3. C. M. BENDER AND S. A. ORSZAG, Advanced Mathematical Methods for Scientists and Engineers, Springer-Verlag, New York, 1999.
4. M. V. BERRY, Uniform asymptotic smoothing of Stokes's discontinuities, Proc. Roy. Soc. London Ser. A, 422 (1989), pp. 7-21.
5. M. V. BERRY, Universal oscillations of high derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), pp. 1735-1751.
6. O. BOKHOVE AND T. G. SHEPHERD, On Hamiltonian balanced dynamics and the slowest invariant manifold, J. Atmospheric Sci., 53 (1996), pp. 276-297.
7. J. P. BOYD, The slow manifold of a five-mode model, J. Atmospheric Sci., 51 (1994), pp. 1057-1064.
8. J. P. BOYD, Eight definitions of the slow manifold: Seiches, pseudoseiches, and exponential smallness, Dyn. Atmos. Oceans, 22 (1995), pp. 49-75.
9. R. CAMASSA, On the geometry of an atmospheric slow manifold, Phys. D, 84 (1995), pp. 357-397.
10. R. CAMASSA AND S.-K. TIN, The global geometry of the slow manifold in the Lorenz-Krishnamurthy model, J. Atmospheric Sci., 53 (1996), pp. 3251-3264.
11. C. J. COTTER, Model Reduction for Shallow Water Dynamics: Balance, Adiabatic Invariance and Subgrid Modelling, Ph.D. thesis, Imperial College London, London, 2004.
12. C. J. COTTER AND S. REICH, Semigeostrophic particle motion and exponentially accurate normal forms, Multiscale Model. Simul., 5 (2006), pp. 476-496.
13. N. FENICHEL, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), pp. 53-98.
14. A. C. FOWLER AND G. KEMBER, The Lorenz-Krishnamurthy slow manifold, J. Atmospheric Sci., 53 (1996), pp. 1433-1437.
15. V. GELFREICH AND L. LEHMAN, Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system, Nonlinearity, 15 (2002), pp. 447-457.
16. V. GELFREICH AND L. LERMAN, Long periodic orbits and invariant tori in a singularly perturbed Hamiltonian system, Phys. D, 176 (2003), pp. 125-146.
17. S. J. JACOBS, Existence of a slow manifold in a model system of equations, J. Atmospheric Sci., 48 (1991), pp. 893-901.
18. H.-O. KREISS, Problems with different time scales for ordinary differential equations, SIAM J. Numer. Anal., 16 (1979), pp. 980-998.
19. H.-O. KREISS AND J. LORENZ, On the existence of slow manifolds for problems with different time scales, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), pp. 159-171.
20. E. N. LORENZ, Attractor sets and quasi-geostrophic equilibrium, J. Atmospheric Sci., 37 (1980), pp. 1685-1699.
21. E. N. LORENZ, On the existence of a slow manifold, J. Atmospheric Sci., 43 (1986), pp. 1547-1557.
22. E. N. LORENZ, The slow manifold-what is it?, J. Atmospheric Sci., 49 (1992), pp. 2449-2451.
23. E. N. LORENZ AND V. KRISHNAMURTHY, On the nonexistence of a slow manifold, J. Atmospheric Sci., 44 (1987), pp. 2940-2950.
24. P. LYNCH, The swinging spring: A simple model for atmospheric balance, in Large-Scale Atmosphere-Ocean Dynamics, Vol. II: Geometric Methods and Models, I. Roulstone and J. Norbury, eds., Cambridge University Press, Cambridge, UK, 2002, pp. 64-108.
25. R. S. MACKAY, Slow manifolds, in Energy Localisation and Transfer, T. Dauxois, A. Litvak-Hinenzon, R. S. MacKay, and A. Spanoudaki, eds., World Scientific, New York, 2004, pp. 149-192.
26. J. RAMIS AND R. SCHÄFKE, Gevrey separation of fast and slow variables, Nonlinearity, 9 (1996), pp. 353-384.
27. R. TEMAM AND D. WIROSOETISNO, Exponential approximations for the primitive equations of the ocean, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), pp. 425-440.
28. N. G. VAN KAMPEN, Elimination of fast variables, Phys. Rep., 124 (1985), pp. 69-160.
29. J. VANNESTE, Inertia-gravity-wave generation by balanced motion: Revisiting the Lorenz-Krishnamurthy model, J. Atmospheric Sci., 61 (2004), pp. 224-234.
30. J. VANNESTE, Wave radiation by balanced motion in a simple model, SIAM J. Appl. Dyn. Syst., 5 (2006), pp. 783-807.
31. J. VANNESTE, Exponential smallness of inertia-gravity-wave, generation at small Rossby number, J. Atmospheric Sci., 65 (2008), pp. 1622-1637.
32. R. VAUTARD AND B. LEGRAS, Invariant manifolds, quasi-geostrophy and initialization, J. Atmospheric Sci., 43 (1986), pp. 565-584.
33. T. WARN, Nonlinear balance and quasi-geostrophic sets, Atmos.-Ocean, 35 (1997), pp. 135-145.
34. T. WARN, O. BOKHOVE, T. G. SHEPHERD, AND G. K. VALLIS, Rossby number expansions, slaving principles, and balance dynamics, Quart. J. R. Met. Soc., 121 (1995), pp. 723-739.
35. D. WIROSOETISNO, Exponentially accurate balance dynamics, Adv. Differential Equations, 9 (2004), pp. 177-196.