Ž . Physics of the Earth and Planetary Interiors 112 1999 21–41 Magnetic field and rotational eigenfrequencies Marianne Greff-Lefftz a, ) , Hilaire Legros b a Institut de Physique du Globe de Paris, Geomagnetism Department, 4 place Jussieu, 75252 Paris 05, France b Institut de Physique du Globe de Strasbourg, 5 rue R. Descartes 67084 Strasbourg, France Received 31 March 1998; revised 18 October 1998; accepted 18 October 1998 Abstract The theory of the rotation of the fluid core is modified in order to take into account the magnetic field within the core. Because of the conductivity of the lower mantle, there is a frictional magnetic torque which appears at the core–mantle Ž . boundary CMB ; the magnitude of this torque depends on the conductivity profile within the mantle and on the magnetic Ž . energy at the CMB. It perturbs the rotational eigenmodes involving a damping in the free core nutation FCN and in the Chandler wobble. The geostrophic pressure at the CMB acts on the bumps of this interface involving a topographic torque. Because of the geostrophic rigidification, this surface pressure field is advected by the core velocity, and consequently, the topography being fixed in a frame related to the mantle, it appears a restoring torque acting on the core. Such a torque perturbs the FCN and creates a slow new rotational eigenmode. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic field; Rotational eigenfrequencies; Core–mantle boundary 1. Introduction The theory of the rotation of a planet having a fluid core has been largely investigated. Its goal is the knowledge of the characteristics of the rotation of the core, of its influence on the Earth’s rotation and in particular on the nutations, and of the transfer mechanism of angular momentum between the core and the mantle. The obtained theoretical results may be compared with precise observed values: observation of the Ž . nutations by V.L.B.I. very long baseline interferometry , observation of the gravity variations using supra-con- ducting gravimeters, observation of the polar motion and of the secular variation of the geomagnetic field. From these comparisons, it appears that the theory of the rotation of the fluid core is in a relatively good agreement with the observations but there remains a discrepancy between the observed and the theoretical value Ž . of the free core nutation FCN and the transfer mechanism of angular momentum between the core and the mantle is not well explained. The theoretical value of the nearly diurnal eigenfrequency is, for the PREM model. w Ž .x w Ž .x yV 1 q 1r458 , where V is the sidereal rotation, whereas the observed value is yV 1 q 1r430 Ž . Defraigne et al., 1994; Florsch et al., 1994 . Two main hypotheses have been proposed to explain this ) Corresponding author. 0031-9201r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž . PII: S0031-9201 98 00154-X