[Étude mathématique du modèle bétaplan : ondes équatoriales et résultats de convergence]
On s’intéresse à un modèle de fluides en rotation rapide, décrivant le mouvement de l’océan dans la zone équatoriale. Ce modèle est connu sous le nom de Saint-Venant, ou système « shallow water », auquel on ajoute un terme de rotation dont l’amplitude est linéaire en la latitude ; en particulier il s’annule à l’équateur. Après une introduction physique au modèle, on décrit les différentes ondes en jeu et l’on étudie en détail les résonances associées à ces ondes. On exhibe ensuite un système limite formel (dans la limite d’une forte rotation), obtenu comme d’habitude en filtrant les ondes, et l’on démontre qu’il est bien posé. Enfin on démontre trois types de résultats de convergence : un théorème de convergence faible vers un système géostrophique linéaire, un théorème de convergence forte des solutions filtrées vers la solution unique du système limite, et enfin un résultat « hybride » de convergence forte des solutions filtrées vers une solution faible du système limite. En particulier on démontre l’absence d’ondes équatoriales confinées dans le mouvement moyen, quand la rotation augmente.
We are interested in a model of rotating fluids, describing the motion of the ocean in the equatorial zone. This model is known as the Saint-Venant, or shallow-water type system, to which a rotation term is added whose amplitude is linear with respect to the latitude; in particular it vanishes at the equator. After a physical introduction to the model, we describe the various waves involved and study in detail the resonances associated to those waves. We then exhibit the formal limit system (as the rotation becomes large), obtained as usual by filtering out the waves, and prove its wellposedness. Finally we prove three types of convergence results: a weak convergence result towards a linear, geostrophic equation, a strong convergence result of the filtered solutions towards the unique strong solution to the limit system, and finally a “hybrid” strong convergence result of the filtered solutions towards a weak solution to the limit system. In particular we obtain that there are no confined equatorial waves in the mean motion as the rotation becomes large.
Keywords: Rotating fluid, betaplane, Kelvin wave, Poincaré wave, Rossby wave, equatorial trapping, weak compactness, filtering method, harmonic analysis
Mot clés : Fluides tournants, Bétaplan, ordre de Kelvin, ordre de Poincaré, ordre de Rossby, confinement équatorial, compacité faible, filtrage, analyse harmonique
@book{MSMF_2006_2_107__1_0,
author = {Gallagher, Isabelle and Saint-Raymond, Laure},
title = {Mathematical study of the betaplane model: {Equatorial} waves and convergence results},
series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
publisher = {Soci\'et\'e math\'ematique de France},
number = {107},
year = {2006},
doi = {10.24033/msmf.419},
mrnumber = {2424189},
zbl = {1151.35070},
language = {en},
url = {http://www.numdam.org/item/MSMF_2006_2_107__1_0/}
}
TY - BOOK AU - Gallagher, Isabelle AU - Saint-Raymond, Laure TI - Mathematical study of the betaplane model: Equatorial waves and convergence results T3 - Mémoires de la Société Mathématique de France PY - 2006 IS - 107 PB - Société mathématique de France UR - http://www.numdam.org/item/MSMF_2006_2_107__1_0/ DO - 10.24033/msmf.419 LA - en ID - MSMF_2006_2_107__1_0 ER -
%0 Book %A Gallagher, Isabelle %A Saint-Raymond, Laure %T Mathematical study of the betaplane model: Equatorial waves and convergence results %S Mémoires de la Société Mathématique de France %D 2006 %N 107 %I Société mathématique de France %U http://www.numdam.org/item/MSMF_2006_2_107__1_0/ %R 10.24033/msmf.419 %G en %F MSMF_2006_2_107__1_0
Gallagher, Isabelle; Saint-Raymond, Laure. Mathematical study of the betaplane model: Equatorial waves and convergence results. Mémoires de la Société Mathématique de France, Série 2, no. 107 (2006), 122 p. doi : 10.24033/msmf.419. http://numdam.org/item/MSMF_2006_2_107__1_0/
[1] – « Un théorème de compacité », C. R. Acad. Sci. Paris 309 (1963), p. 5042–5044. | MR | Zbl
[2] & – « 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. | MR | Zbl
[3] , & – « On some compressible fluid models: Korteweg, lubrication and shallow water systems », Comm. Partial Diff. Equations 28 (2003), p. 843–868. | MR | Zbl
[4] , , & – Basics of Mathematical Geophysics, An introduction to rotating fluids and the Navier-Stokes equations, Oxford Lecture Ser. Math. Appl., vol. 32, University Press, 2006.
[5] – « Zero Mach number limit for compressible flows with periodic boundary conditions », Amer. J. Math. 124 (2002), no. 6, p. 1153–1219. | MR | Zbl
[6] & – « On the homogeneous model of wind-driven ocean circulation », SIAM J. Appl. Math. 60 (2000), no. 1, p. 43–60. | MR | Zbl
[7] & – « The dynamics of equatorial long waves: a singular limit with fast variable coefficients », Commun. Math. Sci. 4 (2006), no. 2, p. 375–397. | MR | Zbl
[8] – « A Remark on smooth solutions of the weakly compressible Navier–Stokes equations », J. Math. Kyoto Univ. 40 (2000), p. 525–540. | MR | Zbl
[9] & – « 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. | MR | Zbl
[11] —, « Weak convergence results for inhomogeneous rotating fluid equations », J. Anal. Math. 99 (2006), p. 1–34. | MR | Zbl
[12] – « Highly rotating fluids in rough domains », J. Math. Pures Appl. 82 (2003), p. 1453–1498. | MR | Zbl
[13] & – « Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation », Discrete Contin. Dynam. Systems (B) 1 (2001), no. 1, p. 89–102. | MR | Zbl
[14] – Atmosphere-Ocean Dynamics, International Geophysics Series, vol. 30, 1982.
[15] & – « Resonant interactions between planetary waves », Proc. Roy. Soc. London A 299 (1967), p. 120–140.
[16] – The theory of rotating fluids, Cambridge monographs on mechanics and applied mathematics, 1969. | MR | Zbl
[17] – « Pseudo-differential energy estimates of singular perturbations », Comm. Pure Appl. Math. 50 (1997), no. 9, p. 821–865. | MR | Zbl
[18] – The Analysis of Linear Partial Differential Equations III, Grundlehren der mathematischen Wissenschaften, vol. 274, Springer Verlag, 1985.
[19] & – « Systematic multi-scale models for the tropics », J. Atmospheric Sci. 60 (2003), p. 173–196.
[20] – Special functions and their applications, Dover Publications, Inc., New York, 1972. | MR
[21] & – Analysis, Graduate Studies in Mathematics, vol. 14, Amer. Math. Soc., 2001. | MR
[22] , & – « New formulations of the primitive equations of atmosphere and applications », Nonlinearity 5 (1992), p. 237–288. | MR | Zbl
[23] – Mathematical Topics in Fluid Mechanics II. Compressible Models, Oxford Science Publications, 1997.
[24] , & – Formulas and Theorems for the Special Functions of Mathematical Physics, Springer Verlag, 1966. | MR | Zbl
[25] – Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes, vol. 9, Amer. Math. Soc., 2003. | MR | Zbl
[26] – « 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
[27] & – « On the barotropic compressible Navier-Stokes equation », to appear in Comm. Partial Diff. Equations. | MR | Zbl
[28] – Geophysical fluid dynamics, Springer, 1979. | Zbl
[29] S. Philander – El Niño, la Niña, and the Southern Oscillation, Academic Press, 1990.
[30] – « Nonlinear wave-wave interactions in a one-layer reduced-gravity model on the equatorial 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] – « Fast singular limits of hyperbolic PDEs », J. Differential Equations 114 (1994), p. 476–512. | MR | Zbl
[34] & – « Some mathematical Problems in Geophysical Fluid Dynamics », in Handbook of Mathematical Fluid Dynamics, vol. III (S. Friedlander & D. Serre, éds.), 2004. | MR | Zbl
[35] (Lord Kelvin) – « On gravitational oscillations of rotating water », Proc. Roy. Soc. Edinburgh 10 (1879), p. 92–100. | JFM
Cité par Sources :
