×

A simple justification of the singular limit for equatorial shallow-water dynamics. (English) Zbl 1156.76013

Summary: The equatorial shallow-water equations at low Froude number form a symmetric hyperbolic system with large variable-coefficient terms. Although such systems are not covered by the classical Klainerman-Majda theory of singular limits, the first two authors recently [A. Dutrifoy and A. Majda, Commun. Partial Differ. Equations 32, No. 10, 1617–1642 (2007; Zbl 1166.35361)] proved that solutions exist uniformly and converge to the solutions of the long-wave equations as the height and Froude number tend to 0. Their proof exploits a special structure of the equations by expanding solutions in series of parabolic cylinder functions. A simpler proof of a slight generalization is presented here in the spirit of the classical theory.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 1166.35361
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Biello, The effect of meridional and vertical shear on the interaction of equatorial baroclinic and barotropic Rossby waves, Stud Appl Math 112 (4) pp 341– (2004) · Zbl 1141.86304
[2] Biello, A new multiscale model for the Madden-Julian oscillation, J Atmospheric Sci 62 (6) pp 1694– (2005)
[3] Bourgeois, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean, SIAM J Math Anal 25 (4) pp 1023– (1994) · Zbl 0811.35097
[4] Chemin, Mathematical geophysics 32 (2006)
[5] Dutrifoy, The dynamics of equatorial long waves: a singular limit with fast variable coefficients, Commun Math Sci 4 (2) pp 375– (2006) · Zbl 1121.35112
[6] Dutrifoy, Fast wave averaging for the equatorial shallow water equations, Comm Partial Differential Equations 32 (110) pp 1617– (2007) · Zbl 1166.35361
[7] Embid, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm Partial Differential Equations 21 (3) pp 619– (1996) · Zbl 0849.35106
[8] Embid, Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers, Geophys Astrophys Fluid Dynam 87 (1) pp 1– (1998)
[9] Frierson, Large scale dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation limit, Commun Math Sci 2 (4) pp 591– (2004) · Zbl 1160.86303
[10] Gallagher, Mathematical study of the betaplane model: equatorial waves and convergence results, Mém Soc Math France (NS) · Zbl 1151.35070
[11] Grenier, Pseudo-differential energy estimates of singular perturbations, Comm Pure Appl Math 50 (9) pp 821– (1997) · Zbl 0884.35183
[12] Joly, Coherent and focusing multidimensional nonlinear geometric optics, Ann Sci École Norm Sup (4) 28 (1) pp 51– (1995) · Zbl 0836.35087
[13] Klainerman, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm Pure Appl Math 34 (4) pp 481– (1981) · Zbl 0476.76068
[14] Klainerman, Compressible and incompressible fluids, Comm Pure Appl Math 35 (5) pp 629– (1982) · Zbl 0478.76091
[15] Majda, Compressible fluid flow and systems of conservation laws in several space variables 53 (1984) · Zbl 0537.76001
[16] Majda, Introduction to PDEs and waves for the atmosphere and ocean 9 (2003) · Zbl 1278.76004
[17] Majda, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J Atmospheric Sci 60 (15) pp 1809– (2003)
[18] Majda, A multiscale model for tropical intraseasonal oscillations, Proc Natl Acad Sci USA 101 (14) pp 4736– (2004) · Zbl 1063.86004
[19] Majda, Averaging over fast gravity waves for geophysical flows with unbalanced initial data, Theoret Comput Fluid Dynam 11 (3) pp 155– (1998) · Zbl 0923.76339
[20] Majda, Systematic multiscale models for the tropics, J Atmospheric Sci 60 pp 393– (2003)
[21] Majda, Interaction of large-scale equatorial waves and dispersion of Kelvin waves through topographic resonances, J Atmospheric Sci 56 (24) pp 4118– (1999)
[22] Philander, El niño, la niña, and the southern oscillation 46 (1990)
[23] Schochet, Fast singular limits of hyperbolic PDEs, J Differential Equations 114 (2) pp 476– (1994) · Zbl 0838.35071
[24] Smith, R. K., ed. The physics and parametrization of moist atmospheric convection. NATO Advanced Study Institute Series C. Mathematical and Physical Sciences, 505. Kluwer, Dordrecht, 1997.
[25] Temam, Handbook of mathematical fluid dynamics pp 535– (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.