FORCE DE LAPLACE EN R ´ EGIME DIFFUSIF 183 et apr`es une transform´ee de Fourier,

− → b = ⁱ − → k .−→ B ηk2 − → v (C.20) La force de Laplace −→ J ×−→ B s’´ecrit alors : − → J ×−→ B ' −→ ∇∧^−→B  ×^−→B +^−→ ∇∧^−→b ^×^−→B (C.21)

et de C.20 on peut obtenir l’effet de la force de Laplace associ´ee aux courants de petite ´ echelle. −→ ∇∧^−→b ^×^−→B = − −→ B .−→ k 2 ηk2 − → v + −→ B .−→ v  −→ B .−→ k  ηk2 − → k (C.22)

qui ne d´epend pas de l’´echelle ! Elle peut ˆetre vue comme une force de freinage magn´etique anisotrope −γ−→

u et une force de pression magn´etique. La force de freinage est exactement celle obtenue par Braginsky (1990). Cette expression montre que

– si −→

B est parall`ele `a −→

v (ce qui implique que −→

B est orthogonal `a −→

k ) il n’y a pas de friction ;

– pour −→

B orthogonal `a−→

v , la pression magn´etique est nulle ; – et pour −→

B orthogonal `a−→

k , la force de Laplace s’annule, mˆeme si−→

v croise des lignes de champ.

Comme nous travaillons avec l’´equation de vorticit´e, nous avons besoin du rotationnel de cette force, qui est simplement

− → ∇∧ −→ B .−→ ∇ −→ b = − −→ B .−→ k 2 ηk2 − → ω (C.23)

Bibliographie

Ashcroft, N. W. & Mermin, N. D. 1976 Solid State Physics. Saunders College.

Aubert, J., Brito, D., Nataf, H.-C., Cardin, P. & Masson, J.-P. 2001 A syste-matic experimental study of spherical shell convection in water and liquid gallium. Phys. Earth Planet. Int. 128, 51–74.

Aubert, J., Gillet, N. & Cardin, P. 2003 Quasigeostrophic models of

convection in rotating spherical shells. Geochem. Geophys. Geosyst. 4(7), 1052 doi :10.1029/2002GC000456.

Baroud, C. N., Plapp, B. B., She, Z.-S. & Swinney, H. L. 2002 Anomalous self-similarity in a turbulent rapidly rotating fluid. Phys. Rev. Letters 88, 114501.

Braginsky, S. 1964 Self excitation of a magnetic field during the motion of a highly conducting fluid. Soviet Phys. JETP 20, 726–735.

Braginsky, S. I. 1990 Local turbulence in the earth’s core. Geophys. Astrophys. Fluid Dynam. 55, 71–87.

Brito, D., Aurnou, J. & Cardin, P. 2004 Turbulent viscosity measurements relevant to planetary core mantle dynamics. Phys. Earth Planetary Interiors 141, 3–8.

Busse, F. H. 1968 Shear flow instabilities in rotating systems. J. Fluid Mech. 33, 577–589.

Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.

Busse, F. H. 1975 A model of geodynamo. Geophys. J. Roy. Astron. 42, 437–459.

Cardin, P. & Brito, D. 2004 Survey of experimental dynamos. in press Presse Univer-sitaire de Grenoble.

Cardin, P., Brito, D., Jault, D., Nataf, H.-C. & Masson, J.-P. 2002 Towards a rapidly rotating liquid sodium dynamo experiment. Magnetohydrodynamics 38, 177–189. Cardin, P. & Olson, P. 1994 Chaotic thermal convection in a rapidly rotating spherical

shell : consequences for flow in the outer core. Phys. Earth Planet. Inter. 82, 235. 185

Christensen, U. 2002 Zonal flow driven by strongly supercritical convection in rotating spherical shells. J. Fluid Mech. 470, 115–133.

Christensen, U., Aubert, J., Cardin, P., Dormy, E., Gibbons, S., Glatzmaier, G., Grote, E., Honkura, Y., Jones, C., Kono, M., Matsushima, M., Saku-raba, A., Takahashi, F., Tilgner, A., Wicht, J. & Zhang, K. 2001 A numerical dynamo benchmark. Phys. Earth Planetary Interiors 128, 25–34.

Christensen, U., Olson, P. & Glatzmaier, G. 1999 Numerical modelling of the geodynamo : a systematic parameter study. Geophys. J. Int. 138, 393.

Danilov, S. & Gurarie, D. 2004 Scaling, spectra and zonal jets in beta-plane turbu-lence. Phys. Fluids 16, 2592–2603.

Dormy, E., Cardin, P. & Jault, D. 1998 MHD flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field. Earth. Planet. Sc. Lett. 160, 15–30.

Dormy, E., Soward, A., Jones, C., Jault, D. & Cardin, P. 2004 The onset of thermal convection in rotating spherical shells. J. Fluid Mech. 501, 43–70.

Dormy, E., Valet, J.-P. & Courtillot, V. 2000 Numerical models of the geodynamo and observational constraints. G-cubed 1, 62.

Dudley, M. & James, R. 1989 Time-dependent kinematic dynamos with stationary flows. Proc. Roy. Soc. Lond. A 425, 407–429.

Frisch, U. 1995 Turbulence, the legacy of A. N. Kolmogorov . Cambridge University Press.

Fr¨uh, W.-G. & Read, P. L. 1999 Experiments on a barotropic rotating shear layer. part 1. instability and steady vortices. J. Fluid Mech. 383, 143–173.

Gailitis, A., Lielausis, O., Platacis, E., Dement´ev, S., Cifersons, A.,

Ger-beth, G., Gundrum, T., Stefani, F., Christen, M. & Will, G. 2001 Magnetic field saturation in the riga dynamo experiment. Phys. Rev. Lett. 86, 3024–3027.

Galperin, B., Sukoriansky, S. & Huang, H.-P. 2001 Universal n⁻⁵ spectrum of

zonal flows on giant planets. Phys. Fluids 13, 1545–1548.

Gillet, N. 2004 Magn´eto-convection dans une sph`ere en rotation rapide : Approches num´erique et exp´erimentale de la convection dans les noyaux plan´etaires. PhD thesis, Universit´e Joseph Fourier.

Glatzmaier, G. & Roberts, P. 1995 A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature 377, 203.

BIBLIOGRAPHIE 187 Gubbins, D., Barber, C. N., Gibbons, S. & Love, J. J. 2000 Kinematic dynamo action in a sphere. i) effects of differential rotation and meridional circulation on solutions with axial dipole symmetry. Proc. R. Soc. Lond. A 456, 1333–1353.

Gubbins, D. & Roberts, P. H. 1987 Geomagnetism. , vol. 2, pp. 1–303. Academic Press.

Hide, R. & Titman, C. W. 1967 Detached shear layers in a rotating fluid. J. Fluid Mech. 29, 39–60.

Hollerbach, R. 2003 Instabilities of the Stewartson layer part 1. The dependence on the sign of Ro. J. Fluid Mech. 492, 289–302.

Jones, C. A., Soward, A. M. & Mussa, A. I. 2000 The onset of thermal convection in a rapidly rotating sphere. J. Fluid Mech. 405, 157–179.

Kageyama, A. & Sato, T. 1997 Generation mechanism of a dipole field by a magneto-hydrodynamic dynamo. Phys. Rev. E 55, 4.

Kiss, A. E. 2003 A modified quasigeostrophic formulation for weakly nonlinear barotropic flow with large-amplitude depth variations. Ocean Model. 5, 171–191.

van de Konijnenberg, J. A., Nielsen, A. H., Rasmussen, J. J. & Stenum, B. 1999 Shear-flow instability in a rotating fluid. J. Fluid Mech. 387, 177–204.

Kono, M. & Roberts, P. 2002 Recent geodynamo simulations and observations of the geomagnetic field. Rev. Geophys. 40, 1013.

Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.

Kuang, W. & Bloxham, J. 1997 A numerical dynamo model in an earth-like dynamical regime. Nature 389, 371–374.

Kuo, H. L. 1949 Dynamic instability of two-dimensional non-divergent flow in a barotropic atmosphere. J. Meteor. 6, 105–122.

L. Mari´e, J. Burguete, F. D. & L´eorat, J. 2003 Numerical study of homogeneous dynamo based on experimental von k´arm´an type flows. Eur. Phys. J. B 33, 469–485. Lesieur, M. 1990 Turbulence in fluids. Kluwer.

Moore, D. W. & Saffman, P. G. 1969 The structure of free vertical shear layers in a rotating fluid and the motion produced by a slowly rising body. Phil. T. R. Soc. Lond. A 264, 597–634.

Niino, H. & Misawa, N. 1984 An experimental and theoretical study of barotropic instability. J. Atmos. Sci. 41, 1992–2011.

Noir, J., Cardin, P., Jault, D. & Masson, J.-P. 2003 Experimental evidence of nonlinear resonance effects between retrograde precession and the tilt-over mode within a spheroid. Geophys. J. Int. 154, 407–416.

Olson, P., Christensen, U. & Glatzmaier, G. A. 1999 Numerical modeling of the geodynamo : Mechanisms of field generation and equilibration. J. Geophys. Res. 104, 10383–10404.

Parker, E. 1955 Hydromagnetic dynamo models. Astrophys. J. 122, 293.

Pedlosky, J. 1987 Geophysical fluid dynamics, 2nd edn. Springer, New York.

Poirier, J.-P. 1994 Physical properties of the earth’s core. C. R. Acad. Sci. Paris 318, 341–350.

Ponty, Y., Pinton, J.-F. & Politano, H. 2004 Simulation of induction at low ma-gnetic prandtl number. Phys. Rev. Letters 92, 144503.

Proudman, I. 1956 The almost-rigid rotation of viscous fluid between concentric spheres. J. Fluid Mech. 1, 505–516.

Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.

Rieutord, M. 1997 Une introduction `a la dynamique des fluides. Masson.

Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for 2-dimensional flows. J. Fluid Mech. 229, 291–310.

Roberts, P. H. 1968 On the thermal instability of a a self-gravitating fluid sphere contai-ning heat sources. Phil. Trans. R. Soc. Lond. A 263, 93–117.

Roberts, P. H. 1972 Kinematic dynamo models. Phil. Trans. R. Soc. Lond. A 272, 663–698.

Roberts, P. H. & Stewartson, K. 1963 On the stability of a Maclaurin spheroid of small viscosity. Astrophys. J. 137, 777–790.

Sarson, G. R. & Busse, F. H. 1998 The kinematic dynamo action of spiralling convec-tive flows. Geophys. J. Int. 133, 140–158.

Schaeffer, N. & Cardin, P. 2005a Quasi-geostrophic kinematic dynamos at low ma-gnetic prandtl number. submitted to Earth Planet. Sci. Lett. ?, ?

Schaeffer, N. & Cardin, P. 2005b Quasi-geostrophic model of the instabilities of the Stewartson layer in flat and depth varying containers. Phys. Fluids ?, ?

BIBLIOGRAPHIE 189 Sommeria, J., Meyers, S. D. & Swinney, H. L. 1988 Laboratory simulation of

Jupi-ter’s Great Red Spot. Nature 361, 689.

Souriau, A., Garcia, R. & Poupinet, G. 2003 The seismological picture of the inner core : structure and rotation. Comptes rendus G´eosciences 335 (1), 51–63.

Soward, A. M. 1977 On the finite amplitude thermal instability of a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dynam. 9, 19–74.

Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 17–26.

Stewartson, K. 1966 On almost rigid rotations. part 2. J. Fluid Mech. 26, 131–144.

Stieglitz, R. & M¨uller, U. 2001 Experimental demonstration of a homogeneous two-scale dynamo. Phys. Fluids 13, 561–564.

Tilgner, A. 2002 Numerical simulation of the onset of dynamo action in an experimental two-scale dynamo. Phys. fluids 14, 4092–4094.

W. Dobler, P. F. & Stepanov, R. 2003 Screw dynamo in a time-dependent pipe flow. Phys. Rev. E 67, 056309.

Yano, J.-I. 1992 Asymptotic theory of thermal convection in rapidly rotating systems. J. Fluid Mech. 243, 103–131.

Yano, J.-I., Talagrand, O. & Drossart, P. 2003 Outer planets : Origins of atmos-pheric zonal winds. Nature 421, 36.

Zavala Sanson, L. & van Heijst, G. J. F. 2002 Ekman effects in a rotating flow over bottom topography. J. Fluid Mech. 471, 239–255.