M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964.

H. Amit, P. Olson, and U. Christensen, Tests of core flow imaging methods with numerical dynamos, Geophysical Journal International, vol.168, issue.1, pp.27-39, 2007.
DOI : 10.1111/j.1365-246X.2006.03175.x

G. Backus, R. L. Parker, and C. Constable, Foundations of Geomagnetism, 1996.

G. E. Backus, Kinematics of Geomagnetic Secular Variation in a Perfectly Conducting Core, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.263, issue.1141, pp.239-266, 1968.
DOI : 10.1098/rsta.1968.0014

S. I. Braginsky, Torsional magnetohydrodynamic vibrations in the Earth's core and variations in day length, Geomag. Aeron, vol.10, pp.1-10, 1970.

B. A. Buffett, J. Mound, and A. Jackson, Inversion of torsional oscillations for the structure and dynamics of Earth's core, Geophysical Journal International, vol.177, issue.3, pp.878-890, 2009.
DOI : 10.1111/j.1365-246X.2009.04129.x

J. Chen, Global mass balance and the length-of-day variation, Journal of Geophysical Research, vol.91, issue.3, pp.10-1029, 2005.
DOI : 10.1029/2004JB003474

U. R. Christensen and J. Aubert, Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields, Geophysical Journal International, vol.166, issue.1, pp.97-114, 2006.
DOI : 10.1111/j.1365-246X.2006.03009.x

URL : https://hal.archives-ouvertes.fr/hal-00149194

U. R. Christensen and J. Wicht, Numerical Dynamo Simulations, Treatise on Geophysics, pp.245-282, 2007.
DOI : 10.1016/b978-044452748-6.00134-6

URL : http://hdl.handle.net/11858/00-001M-0000-0014-FEE0-4

U. R. Christensen, V. Holzwarth, and A. Reiners, Energy flux determines magnetic field strength of planets and stars, Nature, vol.461, issue.7226, pp.167-169, 2009.
DOI : 10.1038/nature07626

P. Courtier, Variational methods, J. of the Meteor. Soc. Jap, vol.75, pp.101-108, 1997.

P. Courtier and O. Talagrand, Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. Ii: Numerical Results, Quarterly Journal of the Royal Meteorological Society, vol.40, issue.478, p.113, 1987.
DOI : 10.1002/qj.49711347813

P. H. Diamond, S. I. Itoh, K. Itoh, and T. S. Hahm, Zonal flows in plasma???a review, Plasma Physics and Controlled Fusion, vol.47, issue.5, pp.35-161, 2005.
DOI : 10.1088/0741-3335/47/5/R01

G. Evensen, Data assimilation: The ensemble Kalman filter, 2007.
DOI : 10.1007/978-3-642-03711-5

C. Eymin and G. Hulot, On core surface flows inferred from satellite magnetic data, Physics of the Earth and Planetary Interiors, vol.152, issue.3, pp.200-220, 2005.
DOI : 10.1016/j.pepi.2005.06.009

URL : https://hal.archives-ouvertes.fr/insu-00372754

A. Fournier, C. Eymin, and T. Alboussì-ere, A case for variational geomagnetic data assimilation: insights from a one-dimensional, nonlinear, and sparsely observed MHD system, Nonlinear Processes in Geophysics, vol.14, issue.2, pp.163-180, 2007.
DOI : 10.5194/npg-14-163-2007

URL : https://hal.archives-ouvertes.fr/hal-00331098

R. Giering and T. Kaminski, Recipes for adjoint code construction, ACM Transactions on Mathematical Software, vol.24, issue.4, pp.437-474, 1998.
DOI : 10.1145/293686.293695

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.7838

J. C. Gilbert and C. Lemaréchal, Some numerical experiments with variable-storage quasi-Newton algorithms, Mathematical Programming, pp.407-435, 1989.
DOI : 10.1007/BF01589113

N. Gillet, M. A. Pais, and D. Jault, Ensemble inversion of time-dependent core flow models, Geochemistry, Geophysics, Geosystems, vol.388, issue.1, pp.10-1029, 2009.
DOI : 10.1029/2008GC002290

URL : https://hal.archives-ouvertes.fr/hal-00377131

G. A. Glatzmaier and P. H. Roberts, A three-dimensional self-consistent computer simulation of a geomagnetic field reversal, Nature, vol.377, issue.6546, pp.203-209, 1995.
DOI : 10.1038/377203a0

R. S. Gross, Earth Rotation Variations -Long Period, Treatise on Geophysics, pp.239-294, 2007.

H. Hersbach, Application of the adjoint of the WAM model to inverse wave modeling, Journal of Geophysical Research: Oceans, vol.29, issue.44, pp.10469-10488, 1998.
DOI : 10.1029/97JC03554

R. Hide, Free Hydromagnetic Oscillations of the Earth's Core and the Theory of the Geomagnetic Secular Variation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.259, issue.1107, pp.615-647, 1966.
DOI : 10.1098/rsta.1966.0026

R. Holme, Large-Scale Flow in the Core, Treatise on Geophysics, pp.107-130, 2007.

R. Holme and N. Olsen, Core surface flow modelling from high-resolution secular variation, Geophysical Journal International, vol.166, issue.2, pp.518-528, 2006.
DOI : 10.1111/j.1365-246X.2006.03033.x

G. Hulot, T. Sabaka, and N. Olsen, The Present Field, Treatise on Geophysics, pp.33-75, 2007.

K. Ide, P. Courtier, M. Ghil, and C. Lorenc, Unified notations for data assimilation: Operational, sequential and variational, J. of the Meteor. Soc. Jap, vol.75, pp.181-189, 1997.

A. Jackson and C. Finlay, Geomagnetic Secular Variation and Its Applications to the Core, Treatise on Geophysics, 2007.
DOI : 10.1016/B978-044452748-6/00090-0

D. Jault, Axial invariance of rapidly varying diffusionless motions in the Earth???s core interior, Physics of the Earth and Planetary Interiors, vol.166, issue.1-2, pp.67-76, 2008.
DOI : 10.1016/j.pepi.2007.11.001

D. Jault, C. Gire, and J. L. , Westward drift, core motions and exchanges of angular momentum between core and mantle, Nature, vol.333, issue.6171, pp.353-356, 1988.
DOI : 10.1038/333353a0

E. Kalnay, M. Kanamitsu, R. Kistler, W. Collins, D. Deaven et al., The NCEP/NCAR 40-Year Reanalysis Project, Bulletin of the American Meteorological Society, vol.77, issue.3, pp.437-471, 1996.
DOI : 10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;2

F. Dimet and O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus A: Dynamic Meteorology and Oceanography, vol.109, issue.2, pp.97-110, 1986.
DOI : 10.3402/tellusa.v38i2.11706

B. Lehnert, Magnetohydrodynamic Waves Under the Action of the Coriolis Force., The Astrophysical Journal, vol.119, pp.647-654, 1954.
DOI : 10.1086/145869

D. Liu, A. Tangborn, and W. Kuang, Observing system simulation experiments in geomagnetic data assimilation, Journal of Geophysical Research, vol.75, issue.3, pp.10-1029, 2007.
DOI : 10.1029/2006JB004691

N. Olsen and M. Mandea, Rapidly changing flows in the Earth???s core, Nature Geoscience, vol.166, issue.6, p.390, 2008.
DOI : 10.1038/ngeo203

N. Olsen, H. Lühr, T. J. Sabaka, M. Mandea, M. Rother et al., CHAOS-a model of the Earth's magnetic field derived from CHAMP, ??rsted, and SAC-C magnetic satellite data, Geophysical Journal International, vol.166, issue.1, pp.67-75, 2006.
DOI : 10.1111/j.1365-246X.2006.02959.x

M. A. Pais and D. Jault, Quasi-geostrophic flows responsible for the secular variation of the Earth's magnetic field, Geophysical Journal International, vol.173, issue.2, pp.421-443, 2008.
DOI : 10.1111/j.1365-246X.2008.03741.x

M. A. Pais, O. Oliveira, and F. Nogueira, Nonuniqueness of inverted core-mantle boundary flows and deviations from tangential geostrophy, Journal of Geophysical Research: Solid Earth, vol.287, issue.B8, pp.10-1029, 2004.
DOI : 10.1029/2004JB003012

E. Plaut, Nonlinear dynamics of traveling waves in rotating Rayleigh-B??nard convection: Effects of the boundary conditions and of the topology, Physical Review E, vol.67, issue.4, p.46303, 2003.
DOI : 10.1103/PhysRevE.67.046303

S. Rau, U. Christensen, A. Jackson, and J. Wicht, Core flow inversion tested with numerical dynamo models, Geophysical Journal International, vol.141, issue.2, pp.485-497, 2000.
DOI : 10.1046/j.1365-246x.2000.00097.x

URL : http://gji.oxfordjournals.org/cgi/content/short/141/2/485

P. H. Roberts, An introduction to magnetohydrodynamics. Longmans, 1967.

Z. Sun, A. Tangborn, and W. Kuang, Data assimilation in a sparsely observed one-dimensional modeled MHD system, Nonlinear Processes in Geophysics, vol.14, issue.2, pp.181-192, 2007.
DOI : 10.5194/npg-14-181-2007

URL : https://hal.archives-ouvertes.fr/hal-00302844

O. Talagrand, Assimilation of observations, an introduction, J. of the Meteor. Soc. Jap, vol.75, pp.191-209, 1997.

O. Talagrand and P. Courtier, Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. I: Theory, Quarterly Journal of the Royal Meteorological Society, vol.8, issue.10, pp.1311-1328, 1987.
DOI : 10.1002/qj.49711347812

S. M. Tobias, P. H. Diamond, and D. W. Hughes, ??-Plane Magnetohydrodynamic Turbulence in the Solar Tachocline, The Astrophysical Journal, vol.667, issue.1, pp.113-116, 2007.
DOI : 10.1086/521978

G. M. Turner, J. L. Rasson, and C. V. Reeves, Observation and Measurement Techniques, Treatise on Geophysics, pp.33-75, 2007.

G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, 2006.
DOI : 10.1017/CBO9780511790447

R. Waddington, D. Gubbins, and N. Barber, Geomagnetic field analysis-V. Determining steady core-surface flows directly from geomagnetic observations, Geophysical Journal International, vol.122, issue.1, pp.326-350, 1995.
DOI : 10.1111/j.1365-246X.1995.tb03556.x

P. Wessel and W. H. Smith, Free software helps map and display data, Eos, Transactions American Geophysical Union, vol.72, issue.41, pp.441-441, 1991.
DOI : 10.1029/90EO00319

S. Zatman and J. Bloxham, Torsional oscillations and the magnetic field within the Earth's core, Nature, vol.388, issue.6644, pp.760-763, 1997.
DOI : 10.1038/41987

S. Zatman and J. Bloxham, in the Earth???s core, Geophysical Journal International, vol.138, issue.3, pp.679-686, 1999.
DOI : 10.1046/j.1365-246x.1999.00909.x