Convergence of the rotating fluids system in a domain with rough boundaries
Journées équations aux dérivées partielles (2003), article no. 8, 15 p.

We consider a rotating fluid in a domain with rough horizontal boundaries. The Rossby number, kinematic viscosity and roughness are supposed of characteristic size ϵ. We prove a convergence theorem on solutions of Navier-Stokes Coriolis equations, as ϵ goes to zero, in the well prepared case. We show in particular that the limit system is a two-dimensional Euler equation with a nonlinear damping term due to boundary layers. We thus generalize the results obtained on flat boundaries with the classical Ekman layers.

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     title = {Convergence of the rotating fluids system in a domain with rough boundaries},
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     eid = {8},
     pages = {1--15},
     publisher = {Universit\'e de Nantes},
     year = {2003},
     doi = {10.5802/jedp.622},
     mrnumber = {2050594},
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     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.622/}
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Gérard-Varet, David. Convergence of the rotating fluids system in a domain with rough boundaries. Journées équations aux dérivées partielles (2003), article  no. 8, 15 p. doi : 10.5802/jedp.622. http://www.numdam.org/articles/10.5802/jedp.622/

[1] Achdou, Y., Pironneau, O., and Valentin, F. Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147, 1 (1998), 187-218. | MR | Zbl

[2] Amirat, Y., Bresch, D., Lemoine, J., and Simon, J. Effect of rugosity on a flow governed by navier-stokes equations. Quarterly Appl. Math 59, 4 (2001), 769-785. | MR | Zbl

[3] Bresch, D., Desjardins, D., and Gérard-Varet, D. Rotating fluids in a cylinder. Prepublication UMPA 317, 2003.

[4] Chemin, J.-Y. Perfect incompressible fluids, vol. 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. | MR | Zbl

[5] Chemin, J.-Y., Desjardins, B., Gallagher, I., and Grenier, E. Ekman boundary layers in rotating fluids. ESAIM Controle optimal et calcul des variations (2002). | EuDML | Numdam | MR | Zbl

[6] Colin, T., and Fabrie, P. Rotating fluid at high Rossby number driven by a surface stress: existence and convergence. Adv. Differential Equations 2, 5 (1997), 715-751. | MR | Zbl

[7] Desjardins, B., Dormy, E., and Grenier, E. Stability of mixed Ekman-Hartmann boundary layers. Nonlinearity 12, 2 (1999), 181-199. | MR | Zbl

[8] Desjardins, B., and Grenier, E. Derivation of quasi-geostrophic potential vorticity equations. Adv. Differential Equations 3, 5 (1998), 715-752. | MR | Zbl

[9] Dormy, E. Modélisation Numérique de la Dynamo Terrestre. PhD thesis, Institut de Physique du Globe de Paris, 1997.

[10] Espedal, M. S., Fasano, A., and Mikelić, A. Filtration in porous media and industrial application, vol. 1734 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2000. Lectures from the 4th C.I.M.E. Session held in Cetraro, August 24-29, 1998, Edited by Fasano, Fondazione C.I.M.E.. [C.I.M.E. Foundation]. | MR

[11] Galdi, G. P. An introduction to the mathematical theory of the Navier-Stokes equations. Vol. II, vol. 39 of Springer Tracts in Natural Philosophy. Springer-Verlag, New York, 1994. Nonlinear steady problems. | MR | Zbl

[12] Gérard-Varet, D. Highly rotating fluids in rough domains. Prepublication UMPA 314, 2003. | MR

[13] Gerard-Varet, D., and Grenier, E. A zoology of boundary layers. Revista de la Real Academia de Ciencias 96, 3 (2002), 401-411. | MR

[14] Greenspan, H. The Theory of Rotating Fluids. Breukelen Press, 1968. | MR | Zbl

[15] Grenier, E., and Masmoudi, N. Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Partial Differential Equations 22, 5-6 (1997), 953-975. | MR | Zbl

[16] Jäger, W., and Mikelić, A. On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differential Equations 170, 1 (2001), 96-122. | MR | Zbl

[17] Marchioro, C., and Pulvirenti, M. Mathematical Theory of Incompressible Nonviscous Fluids. Springer Verlag, 1994. | MR | Zbl

[18] Masmoudi, N. Ekman layers of rotating fluids: the case of general initial data. Comm. Pure Appl. Math. 53, 4 (2000), 432-483. | MR | Zbl

[19] Narteau, C., Le Mouël, J.-L., J.-P. Poirier, J.-P., Sepulveda, E., and Shnirman, M. On a small scale roughness of the core-mantle boundary. Earth and Plan. Science Letters 191 (2001), 49-61.

[20] Pedlosky, J. Geophysical Fluid Dynamics. Springer Verlag, 1979. | Zbl

[21] Rousset, F. Stability of large ekman layers in rotating fluids. submitted, 2003. | MR

[22] Stewartson, K. On almost rigid rotations. J. Fluid Mech. 3 (1957), 17-26. | MR | Zbl

[23] Stewartson, K. On almost rigid rotations. part2. J. Fluid Mech. 26 (1965), 131-152. | MR | Zbl

[24] Temam, R. Navier-Stokes Equations. North-Holland, 1985. | MR | Zbl

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