DO WE NEED VARIOUS ASSUMPTIONS TO GET A GOOD FCN ?

Free core nutation (FCN) is a rotational modes of the Earth with fluid core. All traditional theoretical methods produce FCN period near 460 sidereal days with PREM Earth model, while precise observations (VLBI + SG tides) say it is approximately 430 days. In order to fill this big gap, astronomers and geophysicists give various assumptions, e.g., increasing core-mantle-boundary (CMB) flattening by about 5%, a strong coupling between nutation and geomagnetic field near CMB, viscous coupling, or topographical coupling cross CMB, etc. Do we really need these unproved assumptions? or is it only the problem of these traditional theoretical methods themselves? Earth models (e.g., PREM) provide accurate and robust profiles of physical parameters, like density and Lame parameters, but their radial derivatives, which are also used in all traditional methods to calculate normal modes (e.g., FCN), nutation and tides of non-rigid Earth theoretically, are not so trustable as the parameters themselves. Moreover, the truncation of the expansion series of displacement vector and stress tensor in traditional methods is also of question. A new stratified spectral method is proposed and applied to the computation of normal modes, to avoid these problems. Our primary result of the FCN period is 435±3 sidereal days. 1. KEY QUESTION TO THE STUDY OF FCN Free core nutation (FCN) is a rotational mode of the Earth with fluid outer core (FOC), as the rotating axes of the FOC and that of the mantle do not coincide. The period of FCN reflects and depends on the physical parameters and dynamics of the FOC, mantle and especially that near core-mantle-boundary (CMB). Moreover, FCN influences strongly the retrograde annual (–1yr.) nutation due to their resonance. Therefore, FCN is key parameter in the study of the Earth rotation and the physics of the Earth interior. Almost all calculated FCN periods from traditional theoretical approaches are near 460 days (e.g., Dehant & Defraigne, 1997; Huang et al., 2001; Rogister, 2001; Mathews et al., 2002; Crossley & Rochester, 2014) with a one-dimensional Earth model like PREM (Dziewonski & Anderson, 1981), while modern precise observations from VLBI and superconductivity tidal gravimetry produce FCN period near 430 days with precision of less than one day (e.g., Malkin, 2007; Cui et al., 2014). Their difference of approximate 30 days is significant large comparing with the observation precision. Although some studies discover that the FCN period may vary in the past three decades, its variation is still less than 3 days and much smaller than the 30 days gap. In order to fill this big gap, astronomers and geophysicists give various assumptions. The earliest and easiest assumption is to increase the flattening of core-mantle-boundary (CMB) by about 5% (see e.g., Newburg et al., 1990; Huang, 1999) from that calculated from hydro-static-equilibrium Earth by Clairaut equation and PREM Earth model, i.e., increasing the difference between the polar radii and the equatorial radii by about 500 meters. Another mechanism proposes a strong coupling between nutation and geomagnetic field near CMB. In the original work of Rochester & Smylie (1965), the equatorial components of the electromagnetic torque act on the mantle is calculated rigorously and the electromagnetic damping of the Chandler wobble is first quantitatively investigated, and it is shown that the electromagnetic core-mantle coupling fails by several orders of magnitude either to generate or to damp the Chandler wobble. Following the same way of this work, Buffett et al. (2002) discuss the effect of this coupling on nutation and FCN and show that this coupling can explain the gap of FCN period. However, Huang et al. (2011) argue that, even using the same values of the electro-magnetic parameters near CMB as used by Buffett et al. (2002), the contribution of this coupling to FCN period is one order of magnitude smaller than required. There are also some other assumptions proposed to interpret this big gap of FCN period. They are viscous coupling (e.g., Mathews & Guo, 2005; Buffett, 2010, and topographical coupling (Wu & Wahr,