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Asymptotic analysis of the primitive equations under the small depth assumption. (English) Zbl 1081.35080
Summary: We study the asymptotic behavior of solutions of the primitive equations (PEs) as the depth of the domain goes to zero. We prove that the solutions of the PEs can be expanded as a sum of barotropic flow and baroclinic flow up to a uniformly bounded (in time and space) initial time layer. The barotropic flow is solution of the 2D Navier-Stokes equations with Coriolis force coupled with density. By employing a comparison theorem, the baroclinic flow can be approximated by a quasi-stationary nonlinear GFD-Stokes problem. This article presents a mathematically rigorous justification that the barotropic flow dominates the baroclinic flow in the motion of the atmosphere and ocean.

35Q30Stokes and Navier-Stokes equations
86A05Hydrology, hydrography, oceanography
86A10Meteorology and atmospheric physics
Full Text: DOI
[1] Avrin, J.: Large eigenvalue existence and global regularity results for Navier -- Stokes equations. J. differential equations 127, 365-390 (1996) · Zbl 0863.35075
[2] Foias, C.; Manley, O.; Temam, R.: Modelling of the interaction of small and large eddies in two-dimensional turbulent flows. RAIRO modél. Math. anal. Numér. 22, No. 1, 93-118 (1988) · Zbl 0663.76054
[3] Hale, J. K.; Raugel, G.: Reaction diffusion equations on thin domains. J. math. Pures appl. 71, 33-95 (1992) · Zbl 0840.35044
[4] Hale, J. K.; Raugel, G.: A damped hyperbolic equation on thin domains. Trans. amer. Math. soc. 329, 185-219 (1992) · Zbl 0761.35052
[5] Haltiner, G.; Williams, R.: Numerical weather prediction and dynamic meteorology. (1980)
[6] C. Hu, Finite dimensional behavior of the primitive equations under small depth assumption, preprint. · Zbl 1142.35071
[7] Hu, C.; Temam, R.; Ziane, M.: Regularity results for GFD-Stokes problem and some related linear elliptic pdes in primitive equations. Chinese ann. Math. 23B, 277-292 (2002) · Zbl 1165.86003
[8] Hu, C.; Temam, R.; Ziane, M.: The primitive equations on the large scale ocean under small depth hypothesis. Discrete contin. Dyn. system 9, 97-131 (2003) · Zbl 1048.35082
[9] Iftimie, D.; Raugel, G.: Some results on the Navier -- Stokes equations in thin 3D domain, special issue in celebration of Jack K. Hale’s 70th birthday, part 4 (Atlanta, GA/Lisbon, 1998). J. differential equations 169, 281-331 (2001)
[10] Ladyzhenskaya, O. A.: The mathematical theory of viscous incompressible flow. (1969) · Zbl 0184.52603
[11] Lions, J. L.; Temam, R.; Wang, S.: New formulation of the primitive equations of the atmosphere and applications. Nonlinearity 5, 237-288 (1992) · Zbl 0746.76019
[12] Lions, J. L.; Temam, R.; Wang, S.: On the equations of the large scale ocean. Nonlinearity 5, 1007-1053 (1992) · Zbl 0766.35039
[13] Lions, J. L.; Temam, R.; Wang, S.: Models of the coupled atmosphere and ocean (CAO I). Comput. mech. Adv. 1, 5-54 (1993) · Zbl 0805.76011
[14] Moise, I.; Temam, R.; Ziane, M.: Asymptotic analysis of the Navier -- Stokes equations in thin domains. Topol. methods nonlinear anal. 10, 249-282 (1997) · Zbl 0957.35108
[15] J. Pedlosky, Geophysical Fluid Dynamics, second ed., Springer, New York. · Zbl 0429.76001
[16] G. Raugel, G. Sell, Navier -- Stokes equations on thin 3D domains I: global regularity of spatially periodic conditions, College de France Proceedings, Pitman Research Notes on Mathematical Series, Pitman, New York, London, 1992. · Zbl 0804.35106
[17] Raugel, G.; Sell, G.: Navier -- Stokes equations on thin 3D domains iglobal attractors and global regularity of solutions. J. amer. Math. soc. 6, 503-568 (1993) · Zbl 0787.34039
[18] E. Simonnet, T. Tachim Medjo, R. Temam, Higher order approximation equations for the primitive equations of the ocean, Proceedings of the Erice Conference, June 2003, to appear. · Zbl 1041.86002
[19] E. Simonnet, T. Tachim Medjo, R. Temam, On the order of magnitude of the baroclinic flow in the primitive equations of the ocean, Ann. Math. Pura Appl., 2004, to appear. · Zbl 1104.76084
[20] R. Temam, Infinite dimensional dynamical systems in Mechanics and Physics, Applied Mathematical Science, Vol. 68, second ed., Springer, New York, 1997. · Zbl 0871.35001
[21] Temam, R.: Navier -- Stokes equations, theory and numerical analysis, reprint of the 1984 version. (2001)
[22] Temam, R.; Ziane, M.: Navier -- Stokes equations in three dimensional thin domains with various boundary conditions. Adv. differential equations 1, 499-546 (1996) · Zbl 0864.35083
[23] R. Temam, M. Ziane, Navier -- Stokes equations in thin spherical shells, Optim. Method. Partial Differential Equations, 1996, pp. 281 -- 314, Contemp. Math. 209. · Zbl 0891.35119
[24] R. Temam, M. Ziane, Some mathematical problems in geophysical fluid dynamics, in: Handbook of Mathematical Fluid Dynamics, vol. 3, 2004, 535 -- 657, North-Holland, Amsterdam. · Zbl 1222.35145
[25] Washington, W. M.; Parkinson, C. L.: Introduction to three-dimensional climate modeling. (1986) · Zbl 0655.76003
[26] Ziane, M.: Regularity results for Stokes type systems. Appl. anal. 58, 263-292 (1995) · Zbl 0837.35030