Mathematical study of the betaplane model: Equatorial waves and convergence results  [ Étude mathématique du modèle bétaplan : ondes équatoriales et résultats de convergence ] (2006)


Gallagher, Isabelle; Saint-Raymond, Laure
Mémoires de la Société Mathématique de France, Tome 107 (2006) vi-116 p doi : 10.24033/msmf.419
URL stable : http://www.numdam.org/item?id=MSMF_2006_2_107__1_0

Bibliographie

[1] J.-P. Aubin« Un théorème de compacité », C. R. Acad. Sci. Paris 309 (1963), p. 5042–5044. Zbl 0195.13002 | MR 152860

[2] D. Bresch & B. Desjardins« Existence of global weak solutions for a 2D viscous shallow water equation and convergence to the quasi-geostrophic model », Commun. Math. Phys. 238 (2003), p. 211–223. Zbl 1037.76012 | MR 1989675

[3] D. Bresch, B. Desjardins & C. Lin« On some compressible fluid models: Korteweg, lubrication and shallow water systems », Comm. Partial Diff. Equations 28 (2003), p. 843–868. Zbl 1106.76436 | MR 1978317

[4] J.-Y. Chemin, B. Desjardins, I. Gallagher & E. GrenierBasics of Mathematical Geophysics, An introduction to rotating fluids and the Navier-Stokes equations, Oxford Lecture Ser. Math. Appl., vol. 32, University Press, 2006.

[5] R. Danchin« Zero Mach number limit for compressible flows with periodic boundary conditions », Amer. J. Math. 124 (2002), no. 6, p. 1153–1219. Zbl 1048.35075 | MR 1939784

[6] B. Desjardins & E. Grenier« On the homogeneous model of wind-driven ocean circulation », SIAM J. Appl. Math. 60 (2000), no. 1, p. 43–60. Zbl 0958.76092 | MR 1740834

[7] A. Dutrifoy & A. Majda« The dynamics of equatorial long waves: a singular limit with fast variable coefficients », Commun. Math. Sci. 4 (2006), no. 2, p. 375–397. Zbl 1121.35112 | MR 2219357

[8] I. Gallagher« A Remark on smooth solutions of the weakly compressible Navier–Stokes equations », J. Math. Kyoto Univ. 40 (2000), p. 525–540. Zbl 0997.35050 | MR 1794519

[9] I. Gallagher & L. Saint-Raymond« On the influence of the Earth’s rotation on geophysical flows », to appear in Handbook of Mathematical Fluid Dynamics, Elsevier, (S. Friedlander & D. Serre, eds), 2006.

[10] —, « On pressureless gases driven by a strong inhomogeneous magnetic field », SIAM J. Math. Analysis 36 (2005), no. 4, p. 1159–1176. Zbl 1145.35095 | MR 2139205

[11] —, « Weak convergence results for inhomogeneous rotating fluid equations », J. Anal. Math. 99 (2006), p. 1–34. Zbl 1132.35440 | MR 2279546

[12] D. Gérard-Varet« Highly rotating fluids in rough domains », J. Math. Pures Appl. 82 (2003), p. 1453–1498. Zbl 1033.76008 | MR 2020807

[13] J.-F. Gerbeau & B. Perthame« Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation », Discrete Contin. Dynam. Systems (B) 1 (2001), no. 1, p. 89–102. Zbl 0997.76023 | MR 1821555

[14] A. GillAtmosphere-Ocean Dynamics, International Geophysics Series, vol. 30, 1982.

[15] A. Gill & M. Longuet-Higgins« Resonant interactions between planetary waves », Proc. Roy. Soc. London A 299 (1967), p. 120–140.

[16] H. GreenspanThe theory of rotating fluids, Cambridge monographs on mechanics and applied mathematics, 1969. Zbl 0181.54303 | MR 639897

[17] E. Grenier« Pseudo-differential energy estimates of singular perturbations », Comm. Pure Appl. Math. 50 (1997), no. 9, p. 821–865. Zbl 0884.35183 | MR 1459589

[18] L. HörmanderThe Analysis of Linear Partial Differential Equations III, Grundlehren der mathematischen Wissenschaften, vol. 274, Springer Verlag, 1985.

[19] R. Klein & A. Majda« Systematic multi-scale models for the tropics », J. Atmospheric Sci. 60 (2003), p. 173–196.

[20] N. LebedevSpecial functions and their applications, Dover Publications, Inc., New York, 1972. MR 350075

[21] E. Lieb & M. LossAnalysis, Graduate Studies in Mathematics, vol. 14, Amer. Math. Soc., 2001. MR 1817225

[22] J.-L. Lions, R. Temam & S. Wang« New formulations of the primitive equations of atmosphere and applications », Nonlinearity 5 (1992), p. 237–288. Zbl 0746.76019 | MR 1158375

[23] P.-L. LionsMathematical Topics in Fluid Mechanics II. Compressible Models, Oxford Science Publications, 1997.

[24] W. Magnus, F. Oberhettinger & R. SoniFormulas and Theorems for the Special Functions of Mathematical Physics, Springer Verlag, 1966. Zbl 0143.08502 | MR 232968

[25] A. MajdaIntroduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes, vol. 9, Amer. Math. Soc., 2003. Zbl 1278.76004 | MR 1965452

[26] N. Masmoudi« Some asymptotic problems in fluid mechanics », in Evolution equations and their applications in physical and life sciences (Bad Herrenalb 1998), Lect. Notes Pure Appl. Math., vol. 215, Dekker, New York, 2001, p. 395–404. Zbl 0977.35109

[27] A. Mellet & A. Vasseur« On the barotropic compressible Navier-Stokes equation », to appear in Comm. Partial Diff. Equations. Zbl 1149.35070 | MR 2304156

[28] J. PedloskyGeophysical fluid dynamics, Springer, 1979. Zbl 0429.76001

[29] S. Philander – El Niño, la Niña, and the Southern Oscillation, Academic Press, 1990.

[30] P. Ripa« Nonlinear wave-wave interactions in a one-layer reduced-gravity model on the equatorial β plane », J. Phys. Oceanogr. 12 (1982), no. 1, p. 97–111.

[31] —, « Weak interactions of equatorial waves in a one-layer model. Part I: General properties », J. Phys. Oceanogr. 13 (1983), no. 7, p. 1208–1226.

[32] —, « Weak interactions of equatorial waves in a one-layer model. Part II: Applications », J. Phys. Oceanogr. 13 (1983), no. 7, p. 1227–1240.

[33] S. Schochet« Fast singular limits of hyperbolic PDEs », J. Differential Equations 114 (1994), p. 476–512. Zbl 0838.35071 | MR 1303036

[34] R. Temam & M. Ziane« Some mathematical Problems in Geophysical Fluid Dynamics », in Handbook of Mathematical Fluid Dynamics, vol. III (S. Friedlander & D. Serre, éds.), 2004. Zbl 1222.35145 | MR 2099038

[35] W. Thomson (Lord Kelvin) – « On gravitational oscillations of rotating water », Proc. Roy. Soc. Edinburgh 10 (1879), p. 92–100. Zbl 11.0689.01